Celebratio Mathematica

[1]L. E. Dickson: “The simplest model for illustrating the conic sections,” Amer. Math. Mon. 1 : 8 (August 1894), pp. 261. MR 1513516 article

[2]L. E. Dickson: “On the number of inscriptible regular polygons,” Bull. Am. Math. Soc. 3 : 5 (1894), pp. 123–125. MR 1557312 article

[3]L. E. Dickson: “Biography: Dr. George Bruce Halsted,” Amer. Math. Mon. 1 : 10 (October 1894), pp. 336–340. MR 1513580 article

[4]L. E. Dickson: “Lowest integers representing sides of a right triangle,” Amer. Math. Mon. 1 : 1 (January 1894), pp. 6–11. MR 1513298 article

[5]L. E. Dickson: “The inscription of regular polygons: Chapter I,” Amer. Math. Mon. 1 : 9 (September 1894), pp. 299–301. MR 1513546 article

[6]L. E. Dickson: “The inscription of regular polygons: Chapter II,” Amer. Math. Mon. 1 : 10 (October 1894), pp. 342–345. MR 1513582 article

[7]L. E. Dickson: “The inscription of regular polygons: Chapter III,” Amer. Math. Mon. 1 : 11 (November 1894), pp. 376–377. MR 1513609 article

[8]L. E. Dickson: “The inscription of regular polygons: Chapter IV,” Amer. Math. Mon. 1 : 12 (December 1894), pp. 423–425. MR 1513645 article

[9]L. E. Dickson: “On the inscription of regular polygons,” Ann. Math. 9 : 1–6 (1894–1895), pp. 73–84. MR 1502183 JFM 26.0568.01 article

[10]L. E. Dickson: “A quadratic Cremona transformation defined by a conic,” Amer. Math. Mon. 2 : 7–8 (July–August 1895), pp. 218–221. Also published in Rend. Circ. Mat. Palermo 9:1 (1895). MR 1513859 article

[11]L. E. Dickson: “Cyclic numbers,” Quart. J. Pure Appl. Math. 27 (1895), pp. 366–377. JFM 26.0206.02 article

[12]L. E. Dickson: “Some fallacies of an angle trisector,” Amer. Math. Mon. 2 : 3 (March 1895), pp. 71–72. MR 1513727 article

[13]L. E. Dickson: “A quadratic Cremona transformation defined by a conic,” Rend. Circ. Mat. Palermo 9 : 1 (1895), pp. 256–259. Also published in Amer. Math. Monthly 2:7–8 (1895). JFM 26.0610.01 article

[14]L. E. Dickson: “Gergonne’s pile problem,” Bull. Am. Math. Soc. 1 : 7 (1895), pp. 184–186. MR 1557376 JFM 26.0239.03 article

[15]L. E. Dickson: “The inscription of regular polygons: Chapter V,” Amer. Math. Mon. 2 : 1 (January 1895), pp. 7–9. MR 1513673 article

[16]L. E. Dickson: “The inscription of regular polygons: Chapter VI,” Amer. Math. Mon. 2 : 2 (February 1895), pp. 38–40. MR 1513697 article

[17]L. E. Dickson: “Analytic functions suitable to represent substitutions,” Amer. J. Math. 18 : 3 (July 1896), pp. 210–218. MR 1505712 JFM 27.0105.02 article

[18]L. E. Dickson: The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group. Ph.D. thesis, University of Chicago, 1896. Advised by E. H. Moore. See also articles in Ann. Math. 11:1–6 (1896–1897) and Ann. Math. 11:1–6 (1896–1897). MR 2936778 phdthesis

[19]L. E. Dickson: “The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, I: Analytic representation of substitutions,” Ann. Math. 11 : 1–6 (1896–1897), pp. 161–183. Part II published in Ann. Math. 11:1–6 (1896–1897). See also Dickson’s PhD thesis. MR 1502221 article

[20]L. E. Dickson: “The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, II: Linear groups,” Ann. Math. 11 : 1–6 (1896–1897), pp. 65–120. Part I published in Ann. Math. 11:1–6 (1896–1897). See also Dickson’s PhD thesis. MR 1502214 article

[21]L. E. Dickson: “Systems of continuous and discontinuous simple groups,” Bull. Am. Math. Soc. 3 : 8 (1897), pp. 265–273. MR 1557521 JFM 28.0320.02 article

[22]L. E. Dickson: “Higher irreducible congruences,” Bull. Am. Math. Soc. 3 : 10 (1897), pp. 381–389. MR 1557535 JFM 28.0185.01 article

[23]L. E. Dickson: “The structure of the hypoabelian groups,” Bull. Am. Math. Soc. 4 : 10 (1898), pp. 495–510. MR 1557641 JFM 29.0119.01 article

This paper gives a marked simplification both in the general conceptions and in the detailed developments of the theory of the two hypoabelian groups of Jordan and of the writer’s generalization [1898] to the Galois field of order \( {}2^n \) of the first hypoabelian group. It is important, especially for the generalization, to give these groups an abstract definition independent of the theory of “exposants d’échange,” by means of which Jordan derived them. The crucial point in the simplified treatment lies in the discovery of the explicit relations \begin{align*} & \sum_{i,j}^{1,\dots,m}\alpha_j^{(i)}\delta_j^{(i)} = m, \\ &\sum_{i,j}^{1,\dots,m}\alpha_j^{(i)}\delta_j^{(i)} + \alpha^{\prime}_1 + \beta^{\prime}_1 + \gamma^{\prime}_1 + \delta^{\prime}_1 = m, \end{align*} satisfied by the substitutions of the simple sub-groups \( J \) and \( J_1 \) respectively, but ruling out the remaining substitutions of the total hypoabelian groups \( G \) and \( G_1 \). We may therefore avoid the dependence made in §§274 and 289 upon the last book of the Traité (see §672, page 506).

Basing the investigation upon the groups \( J \) and \( J_1 \) which are to be proved simple, and not upon \( G \) and \( G_1 \) as in the earlier treatments, we wholly avoid the delicate analysis and calculations necessary in §§275 and 290. For the first hypoabelian group, the sub-division into cases is diminished one-half. For the second hypoabelian group, decided simplifications may be made in §§284, 286–8. Some errors have been detected; thus the groups \( G \) and \( G_1 \) do not have the same order, as stated in Jordan, §279. §291 is wholly wrong.

[24]L. E. Dickson: “A new solution of the cubic equation,” Amer. Math. Mon. 5 : 2 (February 1898), pp. 38–39. MR 1514484 article

[25]L. E. Dickson: “Concerning a linear homogeneous group in \( C_{m,\,q} \) variables isomorphic to the general linear homogeneous group in \( m \) variables,” Bull. Am. Math. Soc. 5 : 3 (1898), pp. 120–135. MR 1557673 JFM 29.0119.02 article

[26]L. E. Dickson: “Systems of simple groups derived from the orthogonal group,” Bull. Am. Math. Soc. 4 : 8 (1898), pp. 382–389. An article with the same title (“the complete memoir” according to LED) was also published in California Acad. Proc. 1 (1898). MR 1557622 article

[27]L. E. Dickson: “The quadratic Cremona transformation,” California Acad. Proc. 1 (1898), pp. 13–23. JFM 29.0570.04 article

[28]L. E. Dickson: “Systems of simple groups derived from the orthogonal group,” California Acad. Proc. 1 (1898), pp. 29–40. A shorter article with the same title was published in Bull. Am. Math. Soc. 4:8 (1898). JFM 29.0121.03 article

[29]L. E. Dickson: “A new triply-infinite system of simple groups obtained by a twofold generalization of Jordan’s first hypoabelian group,” Bull. Am. Math. Soc. 5 : 1 (1898), pp. 10–12. Abstract only. Available open access here. JFM 29.0121.04 article

[30]L. E. Dickson: “A triply infinite system of simple groups,” Quart. J. Pure Appl. Math. 29 (1898), pp. 169–178. JFM 28.0136.01 article

[31]L. E. Dickson: “Orthogonal group in a Galois field,” Bull. Am. Math. Soc. 4 : 5 (1898), pp. 196–200. MR 1557585 article

[32]L. E. Dickson: “Determination of the structure of all linear homogeneous groups in a Galois field which are defined by a quadratic invariant,” Amer. J. Math. 21 : 3 (July 1899), pp. 193–256. MR 1505798 JFM 30.0137.03 article

Following the study of certain classes of finite linear groups defined by a quadratic invariant, it seems desirable to have a complete determination of this important type of groups. Besides the work of Jordan [1870, pp. 195–213, 440] on the two hypoabelian groups in the field of integers taken modulo \( {}2 \), and the writer’s generalization [1898a, 1898b] of the first hypoabelian group to the Galois field of order \( {}2^n \), the structures of the orthogonal group [1898c] on \( m \) indices in the Galois field of order \( p^n \) (aside from certain low values of \( m \), \( n \), \( p \)) and of the group [1899] in the same field, leaving invariant the quadratic form \( \sum_{i=1}^m\xi_i\eta_i \), have been previously determined by the writer.

By setting up a complete set of canonical forms for quadratic forms in \( m \) variables in every Galois field, we are able to prove that there exist but two new distinct types of groups defined by a quadratic invariant, one of these being a generalization of the second hypoabelian group of Jordan. Two new systems of simple groups are thus obtained [see §56]. The investigation completes and correlates the results of the earlier papers. It has been the aim throughout to devise methods which require as few separations into cases and special treatments of lower cases as possible. The earlier methods for the orthogonal group have been abandoned in the main.

[33]L. E. Dickson: “The structure of certain linear groups with quadratic invariants,” Proc. London Math. Soc. 30 : 1 (1899), pp. 70–98. MR 1575479 JFM 30.0137.02 article

[34]L. E. Dickson: “Sur plusieurs groupes lineaires isomorphes au groups simple d’ordre \( {}25920 \)” [On several linear groups isomorphic to the simple group of order 25920], C. R. Acad. Sci. Paris 128 (1899), pp. 873–875. JFM 30.0141.02 article

[35]L. E. Dickson: “Sur une generalisation du théorème de Fermat” [On a generalization of Fermat’s theorem], C. R. Acad. Sci. Paris 128 (1899), pp. 1083–1085. French translation of article in Ann. Math. 1:1–4 (1898–1900). JFM 30.0185.02 article

[36]L. E. Dickson: “The largest linear homogeneous group with an invariant Pfaffian,” Bull. Am. Math. Soc. 5 : 3 (1899), pp. 338–342. JFM 30.0142.02 article

In the December number of the Bulletin (pp. 120–135) I have shown that the second compound of the general \( {}2m \)-ary linear homogeneous group is a linear group in \( C_{2m,2} \equiv m(2m-1) \) variables which leaves invariant the Pfaffian \[ F \equiv [1, 2,\dots,2m]. \] Denoting the variables as follows: \begin{equation*}\tag{1} Y_{ij} \equiv -Y_{ji} \qquad (i,j = 1,\dots,2m;\ i\neq j), \end{equation*} the second compound was proved to contain exactly \( (2m)^2 \) linearly independent infinitesimal transformations \begin{equation*}\tag{2} \sum_{r\neq s, t}^{r=1,\dots,2m} Y_{rt}\frac{\partial f}{\partial Y_{rs}}\delta t \qquad (t,s = 1,\dots,2m). \end{equation*} The object of the present note is to prove that the largest linear homogeneous group \( G \) in the \( m(2m-1) \) variables (1) which leaves invariant the Pfaffian \( F \) contains only the \( (2m)^2 \) linearly independent transformations (2).

[37]L. E. Dickson: “The group of linear homogeneous substitutions on \( m_q \) variables which is defined by the invariant \( \Phi=\sum_{i=1}^{i=m}\xi_{i1}\xi_{i2}\dots\xi_{iq} \),” Proc. London Math. Soc. 30 : 1 (1899), pp. 200–208. MR 1575464 JFM 30.0141.01 article

[38]L. E. Dickson: “The known finite simple groups,” Bull. Am. Math. Soc. 5 : 10 (1899), pp. 470–475. JFM 30.0143.01 article

[39]L. E. Dickson: “Report on the recent progress in the theory of linear groups,” Bull. Am. Math. Soc. 6 : 1 (1899), pp. 13–27. JFM 30.0042.01 article

[40]L. E. Dickson: “The first hypoabelian group generalized,” Quart. J. Pure Appl. Math. 30 (1899), pp. 1–16. JFM 29.0118.01 article

[41]L. E. Dickson: “Simplicity of the Abelian group on two pairs of indices in the Galois field of order \( {}2^n \), \( n > 1 \),” Quart. J. Pure Appl. Math. 30 (1899), pp. 383–384. JFM 29.0118.02 article

[42]L. E. Dickson: “Concerning the four known simple linear groups of order \( {}25920 \), with an introduction to the hyper-abelian linear groups,” Proc. London Math. Soc. 31 : 1 (1899), pp. 30–68. MR 1576714 JFM 30.0138.01 article

In a paper [Dickson 1899] giving a résumé of the known systems of simple groups and a table of the orders of all known simple groups not exceeding one million, I find that, apart from the order \( {}25920 \), every case in which two or more simple groups of the same order exist has been completely investigated as to their simple isomorphism or non- isomorphism. The greater part of the present paper deals with the four known simple groups of order \( {}25920 \), viz.,

The simple group \( A(4,3) \), defined by the decomposition of the Abelian group on four indices taken modulo \( {}3 \).
The second hypo-Abelian group \( SH(6,2) \), a sub-group of index \( {}2 \) under the general hypo-Abelian group on six indices taken modulo \( {}2 \).
The orthogonal group \( O(5,3) \), a sub-group of index \( {}4 \) under the total orthogonal group on five indices taken modulo \( {}3 \).
The hyper-Abelian group \( HA(4,2^2) \) of quaternary hyper-Abelian substitutions of determinant unity in the Galois field of \( {}3 \) order \( {}2^2 \).

[43]L. E. Dickson: “The abstract group isomorphic with the symmetric group on \( k \) letters,” Proc. London Math. Soc. 31 : 1 (1899), pp. 351–353. MR 1576717 JFM 30.0141.03 article

[44]L. E. Dickson: “The structure of the linear homogeneous groups defined by the invariant \( \lambda^1\zeta_1^r+\lambda^2\zeta_2^r+\cdots+\lambda^m\zeta_m^r \),” Math. Ann. 52 : 4 (1899), pp. 561–581. MR 1511073 JFM 30.0140.01 article

We study the largest linear homogeneous group in \( m \) variables which leaves absolutely invariant the function \[ \phi_r \equiv \sum_{i=1}^m\lambda_i\xi_i^r \] (each \( \lambda_i\neq 0 \)). For \( r > 2 \), there exists no continuous group leaving \( \phi_r \) invariant; while for \( r=2 \) the continuous group defined is the well known orthogonal group. For \( r > 2 \), every collineation leaving \( \phi_r = 0 \) invariant merely interchanges the terms \( \lambda_i\xi_i^r \).

The greater part of the investigation is concerned with linear substitutions whose coefficients are complexes in the Galois field) of order \( p^n \), including the special case of integers taken modulo \( p \). The case \( r = 2 \) is not considered in the present paper, having been treated at length by the writer in earlier papers [1899, 1898]

[45]L. E. Dickson: “A generalization of Fermat’s theorem,” Ann. Math. (2) 1 : 1–4 (1899–1900), pp. 31–36. French translation published in C. R. Acad. Sci. Paris 128 (1899). MR 1502247 JFM 30.0185.01 article

In a number of investigations, apparently not related to each other, there occurs the following function: \begin{align*} F(a,N) \equiv a^N &- \bigl(a^{N/p_1} + a^{N/p_2} + \cdots + a^{N/p_s}\bigr)\\ &+ \bigl(a^{N/p_1p_2} + a^{N/p_1p_3} + \dots + a^{N/p_{s-1}p_s}\bigr)\\ &- \bigl(a^{N/p_1p_2p_3} + \dots + a^{N/p_{s-2}p_{s-1}p_s}\bigr)\\ &+ \dots + (-1)^sa^{N/p_1p_2\dots p_s}, \end{align*} \( a \) being any integer and \( N \) any positive integer whose distinct prime factors are \( p_1 \), \( p_2,\dots, \) \( p_s \). The theorem which we shall consider in the present paper is that \( F(a,N) \) is divisible by \( N \) for every \( a \) and \( N \). This theorem is a generalization of Fermat’s theorem, to which it reduces when \( N \) is prime.

If §§2–5 of the present paper it is explained how the function \( F(a,N) \) has occured in four distinct mathematical researhes, and how from each of these points of view indirect proofs of the above mentioned generalized theorem have been obtained. In §6 two new direct proofs of this theorem are given. In §7 a third new direct proof is given, based upon a relation observed by Picquet. In §§7 and 8 some further properties of the function \( F(a,N) \) are considered.

[46]L. E. Dickson: “A new definition of the general Abelian linear group,” Trans. Am. Math. Soc. 1 : 1 (1900), pp. 91–96. MR 1500527 JFM 31.0138.04 article

We may give a striking definition of the general Abelian group, making use of the fruitful conception of the “compounds of a given linear homogeneous group,” introduced in recent papers by the writer [1898]. In §3 we determine the multiplicity of the isomorphism of a given linear homogeneous group to its compound groups. This result is applied in §4 to show the simple relation of the Abelian group to the general linear homogeneous group in the same number of variables. In §5 it is shown that the simple groups of composite order which are derived from the decompositions of the quaternary Abelian group and the quinary orthogonal group, each in the \( \mathit{GF}[p^n] \), \( p > 2 \), are simply isomorphic. The investigation affords a proof of the simple isomorphism between the corresponding ten-parameter projective groups without the consideration of their infinitesimal transformations.

[47]L. E. Dickson: “Definition of the abelian, the two hypoabelian, and related linear groups as quotient-groups of the groups of isomorphisms of certain elementary groups,” Trans. Am. Math. Soc. 1 : 1 (1900), pp. 30–38. MR 1500521 JFM 31.0138.03 article

The present paper aims to give a natural definition of the Abelian and two hypoabelian groups, which moreover preserves the essence of Jordan’s definition based upon his important, but artificial, conception of “exposants d’échange” (Traité, pp. 179, 195). A second formal definition, by means of the invariants (11), (14) and (21) below, may be obtained from Jordan (l.c. p. 217 and pp. 438–440).

Following in the main the developments of Jordan (l.c. pp. 420–447, in particular) on the construction of solvable groups, we may obtain the above groups as quotient-groups of the groups of isomorphisms of certain elementary groups. In May, 1898, I communicated such a treatment to Professor Moore, who emphasized the desirability of presenting the definitions thus obtained for the Abelian and hypoabelian groups in two distinct ways: viz., from the standpoint of Jordan’s linear groups and from the standpoint of abstract groups.

Quite recently I have made the investigation from the latter point of view and have found the method so much simpler and more desirable that I have abandoned my earlier work, which was not fully complete, and give in §§8–10 a mere outline of it.

[48]L. E. Dickson: “Proof of the existence of the Galois field of order \( p^r \) for every integer \( r \) and prime number \( p \),” Bull. Am. Math. Soc. 6 : 5 (1900), pp. 203–204. MR 1557697 JFM 31.0142.02 article

Existence proofs have been given by Serret [1879, pp. 122–142] and by Jordan [1870, pp. 16, 17]. The developments used by Serret are lengthy but quite in the spirit of Kronecker’s ideas. The short proof by Jordan, however, assumes with Galois the existence of imaginary roots of an irreducible congruence modulo \( p \).

The proof sketched in this note proceeds by induction. Assuming the existence of the \( \mathit{GF}[p^n] \), we derive that of the \( \mathit{GF}[p^{nq}] \), \( q \) being an arbitrary prime number. Since the \( \mathit{GF}[p] \) exists, being the field of integers taken modulo \( p \), it will follow that the \( \mathit{GF}[p^q] \) exists, and by a simple induction that the \( \mathit{GF}[p^r] \) exists for \( r \) arbitrary.

[49]L. E. Dickson: “A class of linear groups including the Abelian group,” Quart. J. Pure Appl. Math. 31 (1900), pp. 60–66. JFM 30.0142.01 article

[50]L. E. Dickson: “Proof of the non-isomorphism of the simple Abelian group on \( {}2m \) indices and the orthogonal group on \( {}2m+1 \) indices for \( m > 2 \),” Quart. J. Pure Appl. Math. 32 (1900), pp. 42–63. JFM 31.0142.01 article

[51]L. E. Dickson: “Errata: ‘Determination of an abstract simple group of order \( {}2^7\cdot3^6\cdot5\cdot7 \) holoedrically isomorphic with a certain orthogonal group and with a certain hyperabelian group’,” Trans. Am. Math. Soc. 1 : 4 (1900), pp. 509. Errata for article in Trans. Am. Math. Soc. 1:3 (1900). MR 1500436 article

[52]L. E. Dickson: “An abstract simple group of order \( {}25920 \),” Proc. London Math. Soc. 32 : 1 (1900), pp. 3–10. MR 1576228 JFM 31.0143.02 article

[53]L. E. Dickson: “Certain subgroups of the Betti–Mathieu group,” Amer. J. Math. 22 : 1 (January 1900), pp. 49–54. MR 1507867 JFM 31.0142.03 article

[54]L. E. Dickson: “Canonical form of a linear homogeneous substitution in a Galois field,” Amer. J. Math. 22 : 2 (April 1900), pp. 121–137. MR 1507868 JFM 31.0140.02 article

A simple canonical form of the general \( m \)-ary linear homogeneous substitution with integral coefficients taken modulo \( p \), a prime, has been obtained by M. Jordan by a rather lengthy analysis [1870, pp. 114–126]. The method may be readily genearlized to apply to substitutions in an arbitrary Galois field. The present paper, however, gives a short proof by induction of the generalized theorem. That the new method is of practical value in actually reducing a given substitution to its canonical form is illustrated by the example of §3.

In §§4–7, the explicit form of all substitutions in the \( \mathit{GF}[p^n] \) commutative with a given substitution in that field is set up. In particular, the number of such substitutions is deduced, the result being in accord with that of M. Jordan [1870, p. 136] for the case \( n=1 \).

In §8 is given a simple criterion for the conjugacy of two linear homogeneous substitutions in a Galois field.

[55]L. E. Dickson: “Determination of an abstract simple group of order \( {}2^7\cdot3^6\cdot 5\cdot 7 \) holoedrically isomorphic with a certain orthogonal group and with a certain hyperabelian group,” Trans. Am. Math. Soc. 1 : 3 (July 1900), pp. 353–370. Errata published in Trans. Am. Math. Soc. 1:4 (1900). MR 1500543 JFM 31.0140.01 article

[56]L. E. Dickson: “Isomorphism between certain systems of simple linear groups,” Bull. Am. Math. Soc. 6 : 8 (1900), pp. 323–328. MR 1557719 JFM 31.0139.01 article

In an article in the Bulletin for July, 1899, giving the known finite simple groups, I made the conjecture that the simple quaternary hyperorthogonal group \( HO(4,p^{2n}) \) in the \( \mathit{GF}[p^{2n}] \) was isomorphic with the second hypoabelian group \( SH(6,2^n) \), the orthogonal group \( O(6,p^n) \), or the group \( NS(6,p^n) \), according as \( p^n \) is of the form \( {}2^n \), \( {}4l - 1 \), or \( {}4l + 1 \) respectively. For the case [Dickson 1899] \( p^n = 2 \), and for the case [Dickson 1900] \( p^n = 3 \), I have proven the conjecture true by setting up abstract groups holoedrically isomorphic with the linear groups in question. The calculations were necessarily long, so that the method of procedure would scarcely be adapted to the case of general \( p^n \). From the correspondence of generators established in these two special cases, I have been led to the short proof for the general case given in this paper.

[57]L. E. Dickson: “Concerning the cyclic subgroups of the simple group \( G \) of all linear fractional substitutions of determinant unity in two non-homogeneous variables with coefficients in an arbitrary Galois field,” Amer. J. Math. 22 : 3 (1900), pp. 231–252. MR 1505834 article

[58]L. E. Dickson: “Linear substitutions commutative with a given substitution,” Proc. London Math. Soc. 32 : 1 (1900), pp. 165–170. MR 1576218 JFM 31.0140.03 article

[59]L. E. Dickson: “Book review: The elements of the differential and integral calculus,” Bull. Am. Math. Soc. 6 : 8 (1900), pp. 348–351. Book by J. W. A. Young and C. E. Linebarger (Appleton, 1900). MR 1557724 article

Jacob William Albert Young	Related
Charles Elijah Linebarger	Related

[60]L. E. Dickson: “Distribution of the ternary linear homogeneous substitutions in a Galois field into complete sets of conjugate substitutions,” Amer. J. Math. 23 : 1 (January 1901), pp. 37–40. MR 1505848 JFM 32.0134.01 article

The substitutions of the general linear homogeneous group \( G_m \) on \( m \) indices with coefficients in the \( \mathit{GF}[p^n] \) may be classified into complete sets of conjugate substitutions by applying the general theorems given in an earlier paper (American Journal, vol. XXII, pp. 121–137). The classification is based upon the canonical forms of the substitutions of \( G_m \). The former depend upon the characteristic determinants of the substitutions \( (\alpha_{ij}) \), viz.: \begin{align*} \Delta(\lambda) &\equiv \begin{vmatrix} \alpha_{11}-\lambda & \alpha_{12} & \cdots & \alpha_{1m} \\ \alpha_{21} & \alpha_{22}-\lambda & \cdots & \alpha_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m1} & \alpha_{m2} & \cdots & \alpha_{mm} - \lambda \end{vmatrix} \\ &\equiv (-1)^m(\lambda^m - \alpha_1\lambda^{m-1} - \alpha_2\lambda^{m-2}\cdots -\alpha_m). \end{align*} Furthermore, \( G_m \) contains a substitution in whose characteristic determinant the coefficients \( \alpha_1 \), \( \alpha_2,\dots, \) \( \alpha_m \) are arbitrary marks of the \( \mathit{GF}[p^n] \) such that \( \alpha_m\neq 0 \). The required substitution is \[ (\alpha_{ij}) \equiv \begin{pmatrix} \alpha_1 & \alpha_2 & \alpha_3 & \cdots & \alpha_{m-1} & \alpha_m \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \end{pmatrix}. \] The present note considers the case \( m = 3 \). In the article following this, Mr. Putnam treats the case \( m = 4 \).

[61]L. E. Dickson: “On systems of isothermal curves,” Amer. Math. Mon. 8 : 10 (October 1901), pp. 187–192. MR 1515374 JFM 32.0564.02 article

[62]L. E. Dickson: Linear groups with an exposition of the Galois field theory. Sammlung von Lehrbüchern VI. B. G. Teubner (Leipzig and Berlin), 1901. Republished in 1958. JFM 32.0128.01 book

[63]L. E. Dickson: “The alternating group on eight letters and the quaternary linear congruence group modulo two,” Math. Ann. 54 : 4 (1901), pp. 564–569. MR 1511128 JFM 32.0130.01 article

Professor Moore has set up an abstract group which is holoedrically isomorphic with the alternating group on \( k \) letters [1896]. A very elementary proof of this result is given in §1. In an article in the Annalen [1898, particularly pp. 435–436] Prof. Moore applied his theorem to prove that the alternating group \( G_{8!/2} \) on \( {}8 \) letters is holoedrically isomorphic with the group \( L \) of linear homogeneous substitutions on \( {}4 \) indices modulo \( {}2 \). In §2 of this paper is given a much simpler set of substitutions of \( L \) corresponding to the generators of \( G_{8!/2} \). In §3 these relations are inverted, giving the substitutions of \( G_{8!/2} \) which correspoinid to the simplest generators of \( L \). We therefore are able to pass readily from any substitution of either of the isomorphic groups to the corresponding substitution of the other.

[64]L. E. Dickson: “Theory of linear groups in an arbitrary field,” Trans. Am. Math. Soc. 2 : 4 (1901), pp. 363–394. Errata published in Trans. Am. Math. Soc. 3:4 (1902). French translation published in C. R. Acad. Sci. Paris 132 (1901). MR 1500573 JFM 32.0131.03 article

[65]L. E. Dickson: “A class of groups in an arbitrary realm connected with the configuration of the \( {}27 \) lines on a cubic surface,” Quart. J. Pure Appl. Math. 33 (1901), pp. 145–173. JFM 32.0133.01 article

[66]L. E. Dickson: “Théorie des groupes linéaires dans un domaine arbitraire de rationalité” [Theory of linear groups in an arbitrary domain of rationality], C. R. Acad. Sci. Paris 132 (1901), pp. 1547–1548. French translation of article in Trans. Am. Math. Soc. 2:4 (1901). JFM 32.0131.02 article

[67]L. E. Dickson: “Concerning real and complex continuous groups,” Bull. Am. Math. Soc. 7 : 8 (1901), pp. 340–350. MR 1557812 JFM 32.0135.01 article

This paper aims to illustrate certain differences and certain analogies between related real and complex continuous groups. Lie’s theory has been developed chiefly for the latter groups, the modifications necessary for real groups being treated quite briefly.

In §§2–4 are exhibited a real group in \( m \) variables and a real group in \( {}2m \) variables, each of \( m^2 \) parameters, such that the corresponding complex groups are of like structure. In §§5–8, it is shown for \( m = 2 \) that the two real groups have different structures. Of the three proofs given, the first two are analytic and involve little technical knowledge of group theory, while the third group is geometric and gives a better insight into the nature of the question.

In §10, it is illustrated for the case \( m = 2 \) how the general \( m \)-ary linear homogeneous complex continuous group gives rise to an isomorphic \( {}2m \)-ary linear homogeneous real continuous group. Similarly, the complex projective groups lead to groups of birational quadratic transformations.

The investigation has direct contact with the author’s determination [1899] of the structure of the largest group in the \( \mathit{GF}[p^{2n}] \) leaving invariant \[ \xi_1\overline{\xi_1} + \xi_2\overline{\xi_2} + \cdots + \xi_m\overline{\xi_m} ,\] where \( \overline{\xi_i} \) is conjugate to \( \xi_i \) with respect to the \( \mathit{GF}[p^n] \); also with the paper by Moore [1898] on the universal invariant of finite groups of linear substitutions.

[68]L. E. Dickson: “Representation of linear groups as transitive substitution groups,” Amer. J. Math. 23 : 4 (October 1901), pp. 337–377. MR 1505871 JFM 32.0134.03 article

One of the advantages of the study of groups of congruences and of groups of linear substitutions in a general Galois field is the ability to deal with groups of high order as well as with infinite systems of groups by means of analytic formula involving a small number of variables. The study of finite groups defined analytically has led to such distinctive methods and the results have been given such a degree of generality that there appears to be some justification for the attitude of many specialists in the theory of substitution groups towards the analytic theory. It is hoped that the present investigation will be a first step in the direction of a closer union of these branches of group theory.

After giving in §1 a proof of the known theorem on the representation of certain quotient-groups derived from the general linear group as doubly transitive substitution groups, and an outline in §3 of the general method of the paper for representing the more important classes of special linear groups, I take up the orthogonal groups in §§4–16, the abelian linear group in §§17-20, and, finally, the hypoabelian groups in §§21–26. A similar investigation for the hyperorthogonal linear group [1899] will form part of a paper to be presented to the Annalen.

[69]L. E. Dickson: “Concerning the Abelian and related linear groups,” Proc. London Math. Soc. 33 : 1 (1901), pp. 313–325. MR 1575746 JFM 32.0131.01 article

The object of the paper is to determine by elementary methods a classification of the substitutions of the special Abeliau group of quaternary substitutions modulo \( {}3 \), \( SA(4,3) \), into complete sets of conjugates. When, complications are not introduced, the methods are applied to the corresponding group \( SA(4,p^m) \) in the general Galois field; in particular, its substitutions of periods \( {}2 \) and \( {}4 \) are determined (§§2–4). The types of substitutions of period \( {}3 \) in \( SA(4,3^n) \) are determined in §7. In §9 is exhibited a table of the non-conjugate types of substitutions of the group \( SA(4,3) \), and the number of conjugates to each within the group. This group, \( SA(4,3) \), is the group for the trisection of the periods of a hyperelliptic function of four periods. The order of \( SA(4,p^n) \) is \[ p^{4n}(p^{4n}-1)(p^{2n}-1) \] By a more complicated analysis, depending upon the possible forms of the characteristic determinant of an Abelian substitution, the corresponding problem for \( SA(4,p^n) \) has been solved by the writer [1901]. In the present paper, on the other hand, the classification is based upon the periods of the substitutions.

In §10 is determined the structure of a group whose definition is analogous to that of the Abelian group.

[70]L. E. Dickson: “Canonical forms of quaternary abelian substitutions in an arbitrary Galois field,” Trans. Am. Math. Soc. 2 : 2 (1901), pp. 103–138. MR 1500559 JFM 32.0130.02 article

For application to the problem of the distribution of the substitutions of a given group into complete sets of conjugates within the group, a set of canonical forms for its substitutions should have the property that two substitutions are conjugate within the group if, and only if, they are reducible to the same canonical form according to a definite scheme of reduction. In particular, if the canonical form belongs to a higher field than the initial \( \mathit{GF}[p^n] \), the new indices introduced must be conjugate with respect to the initial field.

In the present paper is given a set of canonical forms of quaternary abelian substitutions in the \( \mathit{GF}[p^n] \) such that the canonical forms likewise belong to the special abelian group \( SA(4,p^n) \), the reduction being effected within the group. From them are derived the ultimate canonical forms, not all belonging to the given abelian group. In the former case, the canonical forms depend on the coefficients of the characteristic equation, in the latter case upon its roots.

[71]L. E. Dickson: “The configurations of the \( {}27 \) lines on a cubic surface and the \( {}28 \) bitangents to a quartic curve,” Bull. Am. Math. Soc. 8 : 2 (1901), pp. 63–70. MR 1557844 JFM 32.0492.01 article

After determining four systems of simple groups in an arbitrary domain of rationality which include the four systems of simple continuous groups of Lie, the writer was led to consider the analogous problem for the five isolated simple continuous groups of \( {}14 \), \( {}52 \), \( {}78 \), \( {}133 \), and \( {}248 \) parameters. The groups of \( {}78 \) and \( {}133 \) parameters are related to certain interesting forms of the third and fourth degrees respectively. They suggested the forms \( C \) (§1) and \( Q \) (§3).

It is shown in §1 that the cubic form \( C \) defines the configuration of the \( {}27 \) straight lines on a cubic surface in ordinary space. After proving this result, the writer observed that the formulae remained unaltered if the notation for the variables was chosen to be \( x_i \), \( y_i \), \( z_{ij}\equiv z_{ji} \) (\( i,j = 1,\dots, \) \( {}6 \); \( j\neq i \)) , a notation given by Burkhardt [1893, p. 339]. The notation (1) has been retained in view of the relation with the later sections and to retain uniformity with the notation of a paper [Dickson 1901] on the transformation group defined by the invariant \( C \) for an arbitrary domain of rationality. The group of the configuration of the \( {}27 \) lines on a cubic surface is exhibited in §2. A study of the quartic form \( Q \) and the group of the configuration defined by it is made in §§3–6.

[72]L. E. Dickson: “Errata: ‘The groups of Steiner in problems of contact’,” Trans. Am. Math. Soc. 3 : 4 (October 1902), pp. 500. Errata for article in Trans. Am. Math. Soc. 3:1 (1902). MR 1500454 article

[73]L. E. Dickson: “Errata: ‘Theory of linear groups in an arbitrary field’,” Trans. Am. Math. Soc. 3 : 4 (October 1902), pp. 500. Errata for article in Trans. Am. Math. Soc. 2:4 (1901). MR 1500452 article

[74]L. E. Dickson: “A class of simply transitive linear groups,” Bull. Am. Math. Soc. 8 : 9 (1902), pp. 394–401. MR 1557918 JFM 33.0153.01 article

[75]L. E. Dickson: “A matrix defined by the quaternion group,” Amer. Math. Mon. 9 : 11 (November 1902), pp. 243–248. MR 1515712 JFM 34.0185.02 article

The Quaternion Group was chosen by Weber [1899, vol. 2, pp. 216–218, 125–128] to illustrate his exposition of a portion of Frobenius’s thoery of group-matrices and group-determinants [Dickson 1902a]. In his treatment of the illustrative example, Weber gives no clue to the reader how his results were obtained originally and makes the verification depend upon two laborious compositions of matrices of order eight. The example may, however, be treated very simply by a method which emphasizes certain remarkable properties enjoyed by group-matrices.

A second purpose of this paper is to furnish an instructive example of the theory of group-matrices as generalized for an arbitrary field (Körper, realm of ratioinality) by the writer [1902b]. The canonical formas are essentially differnt in the cases of a field having modulus \( {}2 \) and a field not having modulus \( {}2 \). The methods used are elementary and the paper is entirely independent of those cited. Incidentally, it illustrates a method of factoring important determinants.

[76]L. E. Dickson: College algebra. J. Wiley & Sons (New York), 1902. JFM 33.0188.04 book

[77]L. E. Dickson: “Book review: Éléments de la théorie des nombres,” Bull. Am. Math. Soc. 8 : 6 (1902), pp. 257–259. Book by E. Cahen (Gauthier-Villars, 1900). MR 1557888 article

[78]L. E. Dickson: “Geometric derivation of certain trigonometric formulae,” Amer. Math. Mon. 9 : 2 (February 1902), pp. 36–37. MR 1515484 article

[79]L. E. Dickson: “The order of a certain senary linear group,” Amer. Math. Mon. 9 : 6–7 (June–July 1902), pp. 149–152. MR 1515614 article

In the March number of the Monthly, the writer determined the factors determinant \( D \) of a certain square matrix of order six: \[ \begin{vmatrix} I & \alpha & \beta & \gamma & \delta & \epsilon \\ \beta & I & \alpha & \delta & \epsilon & \gamma \\ \alpha & \beta & I & \epsilon & \gamma & \delta \\ \gamma & \delta & \epsilon & I & \alpha & \beta \\ \delta & \epsilon & \gamma & \beta & I & \alpha \\ \epsilon & \gamma & \delta & \alpha & \beta & I \end{vmatrix} \] It is readily shown that the product of two such matrices is a third matrix of the same form. Hence, if we assign to \( I \), \( \alpha \), \( \beta \), \( \gamma \), \( \delta \), \( \epsilon \) all sets of values in a given field, such that \( D \) does not equal \( {}0 \), we obtain a set of matrices having the group property. The group may be represented concretely as a linear homogeneous group in six variables. It is proposed to determine the order of this group in the Galois Field of order \( p^n \), designated \( \mathit{GF}[p^n] \).

[80]L. E. Dickson: “Theorems on the residues of multinomial coefficients with respect to a prime modulus,” Quart. J. Pure Appl. Math. 33 (1902), pp. 378–384. JFM 33.0203.02 article

[81]L. E. Dickson: “Linear groups in an infinite field,” Proc. London Math. Soc. 34 : 1 (1902), pp. 185–205. MR 1575497 JFM 33.0152.01 article

[82]L. E. Dickson: “On the group defined for any given field by the multiplication table of any given finite group,” Trans. Am. Math. Soc. 3 : 3 (1902), pp. 285–301. MR 1500600 JFM 33.0150.01 article

In two papers [1898], each having the title “On the Continuous Group that is defined by any given Group of Finite Order,” Burnside establishes certain results of decided interest and importance, among them being the theorems of Frobenius [1896] on the irreducible factors of group-determinants. The object of this paper is the development of the theory of analogous groups in any arbitrary field or domain of rationality. In particular, when the field is the general Galois field of order \( p^n \), we obtain a doubly-infinite system of finite groups corresponding to each given finite group. An exceptional case not treated here is that of a field having a modulus which is a factor of the order of the given finite group.

[83]L. E. Dickson: “The groups of Steiner in problems of contact (Second paper),” Trans. Am. Math. Soc. 3 : 3 (July 1902), pp. 377–382. The original article was published in Trans. Am. Math. Soc. 3:1 (1902). MR 1500607 JFM 33.0153.03 article

Denote by \( G \) the group of the equation upon which depends the determination of the curves of order \( n-3 \) having simple contact at \( n(n-3)/2 \) points with a given curve \( C_n \) of order \( n \) having no double points. The case in which \( n \) is odd was discussed in the former paper (Transactions, January, 1902) and \( G \) was shown to be a subgroup of the group defined by the invariants \( \phi_3 \), \( \phi_4 \), \( \phi_5, \dots, \) the latter group being holoedrically isomorphic with the first hypoabelian group on \( {}2p \) indices with coefficients taken modulo \( {}2 \). For \( n \) even, \( G \) is contained in the group \( H \) defined by the invariants \( \phi_4 \), \( \phi_6,\dots, \) with even subscripts. Jordan has shown (Traité, pp. 229–242) that \( H \) is holoedrically isomorphic with the abelian linear group \( A \) on \( {}2p \) indices with coefficients taken modulo \( {}2 \). The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions \( [\alpha_1 \), \( \beta_1,\dots, \) \( \alpha_p \), \( \beta_p] \) of Jordan, and which exhibits explicitly the correspondence between the substitutions of the isomorphic groups.

[84]L. E. Dickson: “The hyperorthogonal groups,” Math. Ann. 55 : 4 (1902), pp. 521–572. MR 1511162 JFM 33.0151.02 article

[85]L. E. Dickson: “Ninth summer meeting of the American Mathematical Society,” Amer. Math. Mon. 9 : 8–9 (August–September 1902), pp. 185–187. MR 1515644 article

[86]L. E. Dickson: “Factors of a certain determinant of order six,” Amer. Math. Mon. 9 : 3 (March 1902), pp. 66–68. MR 1515512 article

[87]L. E. Dickson: “Errata: ‘Canonical forms of quaternary abelian substitutions in an arbitrary Galois field’,” Trans. Am. Math. Soc. 3 : 4 (1902), pp. 499. Errata for article in Trans. Am. Math. Soc. 2:2 (1901). MR 1500448 article

[88]L. E. Dickson: “The groups of Steiner in problems of contact,” Trans. Am. Math. Soc. 3 : 1 (1902), pp. 38–45. A second article with this title was published in Trans. Am. Math. Soc. 3:3 (1902). Errata were published in Trans. Am. Math. Soc. 3:4 (1902). MR 1500585 JFM 33.0153.02 article

The problems of contact discussed by Steiner [1855] and Hesse [1855] were investigated from a more general standpoint by Clebsch in his paper [1864] on the application of Abelian functions to geometry. A study of the groups of these geometrical problems has been made by Jordan [1870, pp. 329–333, pp. 305–308, pp. 229–249]. One of the most interesting of these groups was shown by Jordan to be holoedrically isomorphic with the first hypoabelian linear group, which plays so important a rôle in various geometrical questions and in the problem of the construction of all solvable groups. As the proof (Traité, pp. 229–249) is quite complicated, it seemed to the writer worth while to publish the elementary proof given below of the isomorphism in question. No use will be made of the Jordan substitutions \( [\alpha_1 \), \( \beta_1,\dots, \) \( \alpha_p \), \( \beta_p] \), neither the origin nor the interpretation of which is apparent.

[89]L. E. Dickson: “Book review: Essays on the theory of numbers: I: Continuity and irrational numbers, II: The nature and meaning of numbers,” Bull. Am. Math. Soc. 8 : 6 (1902), pp. 259–260. Book by R. Dedekind (Open Court, 1901). MR 1557889 article

[90]L. E. Dickson: “Canonical form of a linear homogeneous transformation in an arbitrary realm of rationality,” Amer. J. Math. 24 : 2 (April 1902), pp. 101–108. MR 1507871 JFM 33.0151.01 article

In volume XXII of the American Journal of Mathematics, the writer investigated the canonical forms of linear homogeneous transformations in a finite field (necessarily a Galois Field). It was shown that the type of canonical field obtained by M. Jordan for the case of the field of integers taken modulo \( p \) is capable of immediate generalization to an arbitrary Galois Field. Instead of the direct, but lengthy, method of proof employed by M. Jordan, the paper cited gives a proof of induction from a smaller to a greater number of irreducible factors of the characteristic determinant. The latter method leads to a canonical form of linear transformations in an arbitrary field \( F \). The formal process of reduction is the same as in the case of a Galois Field [1900]. But the proof that the reduced form has the desired properties is necessarily more complicated for the general field \( F \) than for a Galois Field. Indeed, the content of the theorem is essentially greater for the general field, since the roots of an equation \( F(K) = 0 \), belonging to and irreducible in \( F \), may not all be rational functions in \( F \) of a single root, as is the case when \( F \) is a Galois Field (see §4). The object of this paper is to give the desired additional proof and thereby establish the validity of the canonical form for an arbitrary field. In §6 is given an example to illustrate the generalized theorem.

[91]L. E. Dickson: “Cyclic subgroups of the simple ternary linear fractional group in a Galois field,” Amer. J. Math. 24 : 1 (January 1902), pp. 1–12. MR 1505877 article

[92]L. E. Dickson: “The known systems of simple groups and their inter-isomorphisms,” pp. 225–229 in Compte rendu du deuxième Congrès International des Mathématiciens (Paris, August 6–12, 1900). Edited by E. Duporcq. Gauthier-Villars (Paris), 1902. JFM 32.0136.01 incollection

[93]L. E. Dickson: “An elementary exposition of Frobenius’s theory of group-characters and group-determinants,” Ann. Math. (2) 4 : 1 (1902), pp. 25–49. MR 1502293 JFM 33.0149.03 article

In a series of recent memoirs [1896a, 1896b, 1897, 1898, 1899] in the Berliner Sitzungsberichte, Frobenius has developed an elaborate theory which has already attracted considerable attention, both on account of the beauty of the results and of its importance in various applications.

The present paper presents the chief results due to Frobenius and follows his methods as closely as an elementary treatment will admit. Symbolic notations are avoided and the material is presented with considerable detail, a number of illustrative examples being worked out. The paper is practically self-contained, involving no dependence upon technical branches of mathematics.

[94]L. E. Dickson: “Three sets of generational relations defining the abstract simple group of order \( {}504 \),” Bull. Am. Math. Soc. 9 : 4 (1903), pp. 194–204. MR 1557975 JFM 34.0166.01 article

[95]L. E. Dickson: “Generational relations defining the abstract simple group of order \( {}660 \),” Bull. Am. Math. Soc. 9 : 4 (1903), pp. 204–206. MR 1557976 JFM 34.0166.02 article

[96]L. E. Dickson: “Definitions of a linear associative algebra by independent postulates,” Trans. Am. Math. Soc. 4 : 1 (1903), pp. 21–26. MR 1500620 JFM 34.0090.02 article

The term linear associative algebra, introduced by Benjamin Peirce, has the same significance as the term system of (higher) complex numbers. In the usual theory of complex numbers, the coördinates are either real numbers or else ordinary complex quantities. To avoid the resulting double phraseology and to attain an evident generalization of the theory, I shall here consider systems of complex numbers whose coördinates belong to an arbitrary field \( F \).

I first give the usual definition by means of a multiplication table for the \( n \) units of the system. It employs three postulates, shown to be independent, relating to \( n^3 \) elements of the field \( F \).

The second definition is of abstract character. It employs four independent postulates which completely define a system of complex numbers.

The first definition may also be presented in the abstract form used for the second, namely, without the explicit use of units. The second definition may also be presented by means of units. Even aside from the difference in the form of their presentation, the two definitions are essentially different.

[97]L. E. Dickson: “On the groups defined for an arbitrary field by the multiplication tables of certain finite groups,” Proc. London Math. Soc. 35 : 1 (1903), pp. 68–80. MR 1577021 JFM 34.0164.01 article

[98]L. E. Dickson: “A generalization of symmetric and skew-symmetric determinants,” Amer. Math. Mon. 10 : 12 (December 1903), pp. 253–256. MR 1516012 article

[99]L. E. Dickson: “Fields whose elements are linear differential expressions,” Bull. Am. Math. Soc. 10 : 1 (1903), pp. 30–31. MR 1558044 JFM 34.0172.01 article

[100]L. E. Dickson: “On the subgroups of order a power of \( p \) in the quaternary abelian group in the Galois field of order \( p^n \),” Trans. Am. Math. Soc. 4 : 4 (October 1903), pp. 371–386. Errata published in Trans. Am. Math. Soc. 5:4 (1904). MR 1500648 JFM 34.0167.03 article

The problem of the \( p \)-section of the periods of hyperelliptic functions of four periods leads to the quaternary abelian group modulo \( p \), where \( p \) is supposed to be an odd prime number. The equation for this \( p \)-section has two essentially distinct resolvents of degree \( (p^4-1)/(p-1) \), as shown by Jordan [1870a, Note E, p. 666; 1870b; 1870c] and as follows incidentally in the present paper, §§2, 4 (Corollary), 16.

For the case \( p=3 \), the group arises in the problem of the \( {}27 \) lines on an general cubic surface, as well as in the reduction of a binary sextic to Cayley’s canonical form \( T^2 - U^3 \).

The question of the existence of resolvents of degree lower than that mentioned above and the related, but more general, problem of the determination of all the subgroups of the abelian group form the subject of investigations now in progress by the writer. On account of the great complexity of these problems only small values of \( p \) are being considered. The discussions for the various values of \( p \) have at least one question in common, that of the subgroups of order a power of \( p \). To avoid duplication, this question is here treated for general \( p \), together with a number of related questions.

[101]L. E. Dickson: “Definitions of a field by independent postulates,” Trans. Am. Math. Soc. 4 : 1 (1903), pp. 13–20. Errata published in Trans. Am. Math. Soc. 5:4 (1904). MR 1500619 JFM 34.0160.02 article

[102]L. E. Dickson: Introduction to the theory of algebraic equations. J. Wiley & Sons (New York), 1903. JFM 34.0210.03 book

[103]L. E. Dickson: “The abstract group \( G \) simply isomorphic with the alternating group on six letters,” Bull. Am. Math. Soc. 9 : 6 (1903), pp. 303–306. MR 1557995 JFM 34.0167.01 article

[104]L. E. Dickson: “The abstract group simply isomorphic with the group of linear fractional transformations in a Galois field,” Proc. London Math. Soc. 35 : 1 (1903), pp. 292–305. MR 1576999 JFM 34.0165.01 article

[105]L. E. Dickson: “On the reducibility of linear groups,” Trans. Am. Math. Soc. 4 : 4 (1903), pp. 434–436. MR 1500653 JFM 34.0168.01 article

The object of this note is a two-fold generalization of Loewy’s theorem proved in these Transactions, vol. 4, pp. 171–177. The generalized theorem is as follows:

Let \( G \) be a group of linear homogeneous transformations with coefficients in a domain \( F \), such that \( G \) is irreducible in \( F \) but is reducible in the domain \( F(\rho_0) \) given by the extension of \( F \) by the adjunction of a root \( \rho_0 \) of an equation belonging to and irreducible in \( F \) and having as its roots \( \rho_0 \), \( \rho_1,\dots, \) \( \rho_{r-1} \). Then \( G \) can be transformed linearly into a decomposable group \[ \begin{array}{ccccc} G_{11} & 0 & 0 & \cdots & 0\\ 0 & G^{\prime}_{11} & 0 & \cdots & 0\\ 0 & 0 & G^{\prime\prime}_{11} & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & G_{11}^{(r-1)} \end{array} \] where \( G_{11}^{(s)} \) is a group irreducible in \( F(\rho_s) \) with coefficients not all in \( F \), and \( G_{11} \), \( G^{\prime}_{11}, \dots, \) \( G_{11}^{(r-1)} \) are conjugate with respect to \( F \).

[106]L. E. Dickson: “Generational relations of an abstract simple group of order \( {}4080 \),” Proc. London Math. Soc. 35 : 1 (1903), pp. 306–319. MR 1577001 JFM 34.0165.02 article

[107]L. E. Dickson: Ternary orthogonal group in a general field. University of Chigago Press, 1903. This monograph and Groups defined for a general field by the rotation groups (1903) were published together in 1904 as a single book. JFM 34.0162.02 book

[108]L. E. Dickson: “Three algebraic notes,” Amer. Math. Mon. 10 : 10 (October 1903), pp. 219–226. MR 1515976 article

[109]L. E. Dickson: Groups defined for a general field by the rotation groups. University of Chigago Press, 1903. This monograph and Ternary orthogonal group in a general field (1903) were published together in 1904 as a single book. JFM 34.0163.01 book

[110]L. E. Dickson: “Generational relations for the abstract group simply isomorphic with the linear fractional group in the \( GF[2^n] \),” Proc. London Math. Soc. 35 : 1 (1903), pp. 443–454. MR 1577013 JFM 34.0167.02 article

[111]L. E. Dickson: “Book review: Niedere Zahlentheorie, erster Teil,” Bull. Am. Math. Soc. 9 : 10 (1903), pp. 555–556. Book by P. Bachmann (Teubner, 1902). MR 1558032 article

[112]L. E. Dickson: “Book review: Liniengeometrie mit Anwendungen,” Bull. Am. Math. Soc. 9 : 10 (1903), pp. 561–562. Book by K. Zindler (Göschen, 1902). MR 1558036 article

[113]L. E. Dickson: “Book review: Éléments de la théorie des groupes abstraits,” Bull. Am. Math. Soc. 11 : 3 (1904), pp. 159–162. Book by J.-A. de Séguier (Gauthier-Villars, 1904). MR 1558181 article

[114]L. E. Dickson: “Book review: An introduction to the modern theory of equations,” Bull. Am. Math. Soc. 11 : 3 (1904), pp. 163–164. Book by F. Cajori (Macmillan, 1904). MR 1558182 article

[115]L. E. Dickson: “Determination of all the subgroups of the known simple group of order \( {}25920 \),” Trans. Am. Math. Soc. 5 : 2 (April 1904), pp. 126–166. Errata were published in Trans. Am. Math. Soc. 5:4 (1904). MR 1500666 JFM 35.0167.02 article

The trisection of the periods of hyperelliptic functions of four periods, the determination of the \( {}27 \) lines on a general cubic surface, and the reduction of a binary sextic to the canonical \( T^2-U^3 \), although apparently unrelated, are not essentially distinct problems from the standpoint of group-theory, since each is readily reduced to the solution of an algebraic equation whose Galois group is the same simple group of order \( {}25920 \). This equation has been shown to possess resolvents of degrees \( {}27 \), \( {}36 \), \( {}40 \) (two essentially distinct ones), and \( {}45 \), but none of degree \( {} < 27 \). The last result was established by Jordan [1870] by an elaborate discussion based on Galois’ theory of algebraic equations. This result is reëstablished in the present paper, which employs only pure group-theory. All the results mentioned follow from the fundamental theorem (not stated heretofore) that all the maximal subgroups of the simple \( G_{25920} \) are conjugate with \( G_{960} \), \( G_{720} \), \( G_{648} \), \( H_{648} \), or \( G_{576} \) (§70). These five groups, appearing in different notations, play a fundamental rôle in the memoirs of Witting and Burkhardt on the geometric and function-theoretic phases of the subject.

[116]L. E. Dickson: “Two systems of subgroups of the quaternary Abelian group in a general Galois field,” Bull. Am. Math. Soc. 10 : 4 (1904), pp. 178–184. MR 1558086 JFM 35.0169.02 article

[117]L. E. Dickson: 1: Ternary orthogonal groups in a general field. 2: The groups defined for a general field by the rotation group. University of Chigago Press, 1904. Combined republication of the monographs Ternary orthogonal group in a general field (1903) and Groups defined for a general field by the rotation groups (1903). JFM 35.0181.04 book

[118]L. E. Dickson: “The subgroups of order a power of \( {}2 \) of the simple quinary orthogonal group in the Galois field of order \( p^n=8l\pm 3 \),” Trans. Am. Math. Soc. 5 : 1 (January 1904), pp. 1–38. Errata were published in Trans. Am. Math. Soc. 5:4 (1904). MR 1500658 JFM 35.0167.01 article

[119]L. E. Dickson: “On the minimum degree of resolvents for the \( p \)-section of the periods of hyperelliptic functions of four periods,” Deutsche Math.-Ver. 13 (1904), pp. 559–560. Available open access here. JFM 35.0166.02 article

[120]L. E. Dickson: “Book review: Funktionentheoretische Vorlesungen,” Bull. Am. Math. Soc. 10 : 6 (1904), pp. 317–321. Book by H. Burkhardt (Veit and Co., 1903). MR 1558115 article

[121]L. E. Dickson: “Addition to the paper on the four known simple linear groups of order \( {}25920 \),” Proc. London Math. Soc. 1 : 1 (1904), pp. 283–284. MR 1576780 JFM 35.0168.01 article

[122]L. E. Dickson: “Application of groups to a complex problem in arrangements,” Ann. Math. (2) 6 : 1 (October 1904), pp. 31–44. MR 1503507 JFM 35.0168.02 article

[123]L. E. Dickson: “A new extension of Dirichlet’s theorem on prime numbers,” Messenger of Mathematics 38 (1904), pp. 155–161. Available open access here. JFM 35.0204.03 article

[124]L. E. Dickson: “Errata: ‘On the subgroups of order a power of \( p \) in the quaternary abelian group in the Galois field of order \( p^n \)’,” Trans. Am. Math. Soc. 5 : 4 (1904), pp. 550–551. Errata for article in Trans. Am. Math. Soc. 4:4 (1903). MR 1500465 article

[125]L. E. Dickson: “Errata: ‘Definitions of a field by independent postulates’,” Trans. Am. Math. Soc. 5 : 4 (1904), pp. 549–550. Errata for article in Trans. Am. Math. Soc. 4:1 (1903). MR 1500461 article

[126]L. E. Dickson: “Errata: ‘The subgroups of order a power of \( {}2 \) of the simple quinary orthogonal group in the Galois field of order \( p^n=8l\pm3 \)’,” Trans. Am. Math. Soc. 5 : 4 (1904), pp. 551. Errata for paper published in Trans. Am. Math. Soc. 5:1 (1904). MR 1500468 article

[127]L. E. Dickson: “Errata: ‘Determination of all the subgroups of the known simple group of order \( {}25920 \)’,” Trans. Am. Math. Soc. 5 : 4 (1904), pp. 551. Errata for article published in Trans. Am. Math. Soc. 5:2 (1904). MR 1500469 article

[128]L. E. Dickson: “Determination of all groups of binary linear substitutions with integral coefficients taken modulo \( {}3 \) and of determinant unity,” Ann. Math. (2) 5 : 3 (1904), pp. 140–144. MR 1503536 JFM 35.0169.01 article

[129]L. E. Dickson: “Book review: Vorlesungen über Algebra,” Bull. Am. Math. Soc. 10 : 5 (1904), pp. 257–260. Book by G. Bauer (Teubner, 1903). MR 1558104 article

[130]L. E. Dickson: “Memoir on Abelian transformations,” Amer. J. Math. 26 : 3 (July 1904), pp. 243–318. MR 1507875 JFM 35.0166.01 article

[131]L. E. Dickson: “A property of the group \( G_2^{2n} \) all of whose operators except identity are of period \( {}2 \),” Amer. Math. Mon. 11 : 11 (November 1904), pp. 203–206. MR 1516217 JFM 35.0181.03 article

[132]L. E. Dickson: “On the subgroups of order a power of \( p \) in the linear homogeneous and fractional groups in the \( \mathit{GF}[p^n] \),” Bull. Am. Math. Soc. 10 : 8 (1904), pp. 385–397. MR 1558132 JFM 35.0170.01 article

[133]L. E. Dickson: “The minimum degree \( \tau \) of resolvents for the \( p \)-section of the periods of hyperelliptic functions of four periods,” Trans. Am. Math. Soc. 6 : 1 (January 1905), pp. 48–57. MR 1500693 JFM 36.0508.06 article

The chief object of the investigation is to prove that, if \( p > 3 \), \[ \tau = (p^4 - 1)/(p - 1). \] The case \( p=3 \) alone is exceptional, the problem then being equivalent to that of the \( {}27 \) lines on a general cubic surface. On the final page of his Traité, Jordan states that he had established the theorem for \( p=5 \), by methods analogous to those used in his complicated discussion for \( p=3 \), and says “mais içi la complication est beaucoup plus grande.”

It is rather remarkable that the minimum \( \tau \) should be so large as \[ p^3 + p^2 + p + 1 ,\] since the fractional form of the general quaternary linear group modulo \( p \) can be represented as a substitution group of this degree (and of no lower in view of the present theorem).

The paper makes considerable headway in the problem of all the subgroups of the quaternary abelian group modulo \( p \), which plays the same rôle in the hyperelliptic modular theory (as yet but little developed) as the binary congruence group plays in the classic elliptic modular theory.

[134]L. E. Dickson: “Determination of the ternary modular groups,” Amer. J. Math. 27 : 2 (April 1905), pp. 189–202. MR 1505964 JFM 36.0208.01 article

The determination of all groups of linear homogeneous transformations on \( m \) variables with coefficients in the \( \mathit{GF}[p^n] \) falls naturally into two cases:

order a multiple of \( p \);
order prime to \( p \).

In the second case, the canonical form of any transformation merely multiplies each variable by a constant, and the problem is analogous to that of the determination of the finite groups of collineations in \( m \) variables. This separation of cases was followed in the treatment of binary groups.

In his elaborate memoir on ternary groups, Burnside [1895] makes the limitation that \( p^2 + p + 1 \) shall be the product of at most two prime factors \( {} > 3 \) or else the triple of such a product. His discussion is occasionally incorrect. In particular, he misses [1895, p. 81] the groups with an invariant ternary quadratic form.

The present paper on the ternary groups of order multiple of \( p \) employs methods entirely different from those used by Burnside. There is no limitation on the odd prime \( p \). Moreover, a representative of each set of conjugate subgroups is exhibited in explicit form.

[135]L. E. Dickson: “On hypercomplex number systems,” Trans. Am. Math. Soc. 6 : 3 (1905), pp. 344–348. MR 1500716 JFM 36.0139.02 article

[136]L. E. Dickson: “On finite algebras,” Nachr. Ges. Wiss. Göttingen (1905), pp. 358–393. JFM 36.0138.03 article

[137]L. E. Dickson: “On the class of the substitutions of various linear groups,” Bull. Am. Math. Soc. 11 : 8 (1905), pp. 426–432. MR 1558235 JFM 36.0210.02 Zbl 0960.14005 article

[138]L. E. Dickson: “Graphical methods in trigonometry,” Amer. Math. Mon. 12 : 6–7 (June–July 1905), pp. 129–133. MR 1516377 article

Aside from the important work on the solution of triangles by diagrams drawn to scale, graphic methods are not usually employed in trigonometry. Even if the cartesian graphs of the trigonometric functions are constructed, no serious applications are made of these graphs. They are, however, admirably adapted to the explanation of interpolation, to the visualization and retention in the memory of the ratios for the angles \( {}0^{\circ} \), \( {}90^{\circ} \), etc. (in contrast to their derivation as limiting values), and to the natural solution of trigonometri equations, — in particular, to the visualization of the number of angles \( {} < 180^{\circ} \) having a given sine or cosine. In addition to these minor advantages resulting from a frequent appeal to the graphs, the graphic method may be employed to perform the highly important service of leading the student naturally to the majority of the fundamental trigonometric formulae, including the addition theorem and formulae for conversion of sum into product. This is in marked constrast to the current method by which each formula makes its appearance from some unseen source, to be followed by a more or less artificial proof.

[139]L. E. Dickson: “A general theorem on algebraic numbers,” Bull. Am. Math. Soc. 11 : 9 (1905), pp. 482–486. MR 1558248 JFM 36.0139.03 article

[140]L. E. Dickson: “Errata: ‘Definitions of a group and a field by independent postulates’,” Trans. Am. Math. Soc. 6 : 4 (October 1905), pp. 547. Errata for an article in Trans. Am. Math. Soc. 6:2 (1905). MR 1500477 article

[141]L. E. Dickson: “On the real elements of certain classes of geometrical configurations,” Ann. Math. (2) 6 : 4 (1905), pp. 141–150. MR 1503554 JFM 36.0209.01 article

[142]L. E. Dickson: “Expressions for the elements of a determinant in terms of the minors of a given order. Generalization of a theorem due to Studnicka,” Amer. Math. Mon. 12 : 12 (December 1905), pp. 217–221. MR 1516491 article

[143]L. E. Dickson: “The group of a tactical configuration,” Bull. Am. Math. Soc. 11 : 4 (1905), pp. 177–179. MR 1558199 JFM 36.0210.01 article

[144]L. E. Dickson: “Determination of all the subgroups of the three highest powers of \( p \) in the group \( G \) of all \( m \)-ary linear homogeneous transformations modulo \( p \),” Quart. J. Pure Appl. Math. 36 (1905), pp. 373–384. JFM 36.0208.03 article

[145]L. E. Dickson: “Book review: Introduction à la théorie des fonctions d’une variable,” Bull. Am. Math. Soc. 11 : 10 (1905), pp. 557–559. Book by J. Tannery (Hermann, 1904). MR 1558260 article

[146]L. E. Dickson: “A new system of simple groups,” Math. Ann. 60 : 1 (1905), pp. 137–150. JFM 36.0206.01 article

One of the five isolated simple continuous groups not occurring in Lie’s four systems is the group of \( {}14 \) parameters studied by Killing, Cartan, and Engel. This group is a special case of a linear group on \( {}7 \) variables with coefficients in an arbitrary field or domain of rationality. The structure of the latter has been determined [1901] by the writer for fields not having modulus \( {}2 \). The problem for modulus \( {}2 \), which requires different analysis, is solved in the present paper. For \( q > 1 \), we obtain a simple group of order \[ 2^{6q}(2^{6q}-1)(2^{2q}-1) .\] For \( q=1 \), the group has a simple subgroup of index \( {}2 \) and order \( {}6048 \). The latter is shown to be holoedrically isomorphic with the simple group [1899] of all ternary hyperorthogonal substitutions of determinant unity in the Galois field of order \( {}3^2 \). The generational relations of the isomorphic abstract group are determined and a transitive representation on \( {}36 \) letters exhibited.

For \( q=1 \), the group of order \( {}12096 \) is shown to be simply isomorphic with a subgroup of index \( {}120 \) of the senary Abelian group modulus \( {}2 \), of order \( {}2^9\cdot 3^4\cdot 5 \cdot 7 \). The latter is known [Jordan 1870, pp. 229–242; Dickson 1902, pp. 377–382] to be simply isomorphic with the group of the equation for the \( {}28 \) bitangents to a quartic curve without double points. It therefore has resolvents of degrees \( {}63 = 2^6-1 \) and \( {}120 \), the latter not hitherto noticed.

[147]L. E. Dickson: “Subgroups of order a power of \( p \) in the general and special \( m \)-ary linear homogeneous groups in the \( \mathit{GF}[p^n] \),” Amer. J. Math. 27 : 3 (July 1905), pp. 280–302. MR 1507878 JFM 36.0208.02 article

It would seem that the most effective method of determining all the subgroups of order a multiple of \( p \) of a linear group in the Galois field of order \( p^n \) is that based upon a complete knowledge of the subgroups of order a power of \( p \). This method has proved successful for the ternary groups [1905] and, as I will show on another occasion, also for the quaternary groups.

The present investigation proceeds far enough to give a clear insight into the nature of the simple laws pervading the subject. It is hoped that the results are capable of extension by induction to all powers of \( p \). To indicate the difficulty of this step, it may be remarked that its completion would give the means of deriving at once an explicit list of all groups of order a power of a prime, and simultaneously all the subgroups of each.

A second aim of the paper was to furnish data for the problem of the determination of all \( m \)-ary groups for low values of \( m \).

[148]L. E. Dickson: “On semi-groups and the general isomorphism between infinite groups,” Trans. Am. Math. Soc. 6 : 2 (1905), pp. 205–208. MR 1500707 JFM 36.0207.02 article

When there exists a correspondence between the elements of two finite groups such that the product of two elements of one corresponds to the product of the corresponding elements of the other, then the elements of either which correspond to the identity of the other group form themselves a group. It is somewhat surprising that this familiar theorem fails in general for infinite groups, as shown by definite examples in §§7–8. Nevertheless, De Séguier [1904] has attempted to establish the theorem for any groups; the error in his argument is quite subtle. The correct theorem involves the concept semi-group, which reduces to a group when there is a finite number of elements, but not in general for an infinitude of elements.

[149]L. E. Dickson: “Definitions of a group and a field by independent postulates,” Trans. Am. Math. Soc. 6 : 2 (1905), pp. 198–204. Errata published in Trans. Am. Math. Soc. textbf{6}:4 (1905). MR 1500706 JFM 36.0207.01 article

[150]L. E. Dickson: “On the cyclotomic function,” Amer. Math. Mon. 12 : 4 (April 1905), pp. 86–89. MR 1516323 article

[151]L. E. Dickson: “Linear algebras in which division is always uniquely possible,” Trans. Am. Math. Soc. 7 : 3 (1906), pp. 370–390. MR 1500755 JFM 37.0111.06 article

[152]L. E. Dickson: “On commutative linear algebras in which division is always uniquely possible,” Trans. Am. Math. Soc. 7 : 4 (1906), pp. 514–522. MR 1500764 JFM 37.0112.01 article

We consider commutative linear algebras in \( {}2n \) units, with coördinates in a general field \( F \), such that \( n \) of the units define a sub-algebra forming a field \( F(J) \). The elements of the algebra may be exhibited compactly in the form \( A + BI \), where \( A \) and \( B \) range over \( F(J) \). As multiplication is not associative in general, \( A \) and \( B \) do not play the rôle of coördinates, so that the algebra is not binary in the usual significance of the term. Nevertheless, by the use of this binary notation, we may exhibit in a very luminous form the multiplication-tables of certain algebras in four and six units, given in an earlier paper [1906]. Proof of the existence of the algebras and of the uniqueness of division now presents no difficulty. The form of the corresponding algebra in \( {}2n \) units becomes obvious. After thus perfecting and extending known results, we attack the problem of the determination of all algebras with the prescribed properties. An extensive new class of algebras is obtained.

[153]L. E. Dickson: “On linear algebras,” Amer. Math. Mon. 13 : 11 (November 1906), pp. 201–205. MR 1516696 JFM 37.0115.03 article

[154]L. E. Dickson: “On the quaternary linear homogeneous groups modulo \( p \) of order a multiple of \( p \),” Amer. J. Math. 28 : 1 (1906), pp. 1–16. MR 1505980 JFM 37.0173.01 article

[155]L. E. Dickson: “The abstract form of the special linear homogeneous group in an arbitrary field,” Quart. J. Pure Appl. Math. 38 (1906), pp. 141–145. JFM 38.0181.01 article

[156]L. E. Dickson: “The abstract form of the Abelian linear groups,” Quart. J. Pure Appl. Math. 38 (1906), pp. 145–158. JFM 38.0181.02 article

[157]L. E. Dickson: “On the theory of equations in a modular field,” Bull. Am. Math. Soc. 13 : 1 (1906), pp. 8–10. MR 1558391 JFM 37.0173.02 article

[158]L. E. Dickson: “Criteria for the irreducibility of functions in a finite field,” Bull. Am. Math. Soc. 13 : 1 (1906), pp. 1–8. MR 1558390 JFM 37.0094.01 article

[159]L. E. Dickson: “On quadratic, Hermitian and bilinear forms,” Trans. Am. Math. Soc. 7 : 2 (1906), pp. 275–292. MR 1500749 JFM 37.0137.01 article

[160]L. E. Dickson: “Modular theory of group characters,” Bull. Am. Math. Soc. 13 : 10 (1907), pp. 477–488. MR 1558503 JFM 38.0180.01 article

The problem of the representation of a given finite group as a linear homogeneous group with real or complex coefficients has been fully treated by Frobenius by means of his theory of group characters. The present paper and the companion paper to appear simultaneously in the Transactions give a first attack on the corresponding problem for linear congruence groups, and in general for finite linear groups in any field \( F \) having a prime modulus \( p \). To obtain simple results, it is in general necessary to introduce certain irrationalities, viz., roots of equations with coefficients in \( F \). As our reference field we shall take the field \( F \) composed of the totality of integral rational functions with integral coefficients of all Galois imaginaries of all degrees, i.e., the roots of congruences irreducible modulo \( p \). In other words, \( F_p \) is the aggregate of the Galois fields \( \mathit{GF}[p^n] \), \( n = 1 \), \( {}2 \), \( {}3, \dots \). Hence every equation with coefficients in \( F_p \) is completely solvable in \( F_p \).

[161]L. E. Dickson: “On quadratic forms in a general field,” Bull. Am. Math. Soc. 14 : 3 (1907), pp. 108–115. MR 1558550 JFM 38.0182.02 article

[162]L. E. Dickson: “Invariants of binary forms under modular transformations,” Trans. Am. Math. Soc. 8 : 2 (1907), pp. 205–232. Errata published in Trans. Am. Math. Soc. 8:4 (1907). MR 1500782 JFM 38.0147.02 article

For a non-modular field \( F \) and a binary form \( f \) with coefficients in \( F \), the problem of the determination of functions of the coefficients and variables of \( f \), invariant under all binary linear transformations in \( F \), is formally identical with the corresponding problem of the ordinary algebraic invariant theory. But for a finite modular field the problem is essentially different; the terms of an invariant need not be of the same degree nor of constant weight; the annihilators are quite complicated, involving higher partial derivatives. Fortunately, the difficulty in the direct computation of the invariants is in marked contrast with the regularity observed in the actual form of the invariants and with the simplicity of the relations between the invariants, common to the algebraic and modular theories, and the additional invariants peculiar to the modular theory.

In the study of the invariants of a given quantic in the Galois field of order \( p^n \), we have a doubly infinite system of problems, corresponding to a single problem in the algebraic theory. Interest naturally centers in a comparative study, rather than in the individual problems. The aim of the present paper is to give in correlation the results of a rather extensive comparative study. Formal proofs of the laws observed are given only in a few instances. It is hoped that proofs of the remaining properties observed will become more practicable when a satisfactory symbolic treatment is constructed.

[163]L. E. Dickson: “Modular theory of group-matrices,” Trans. Am. Math. Soc. 8 : 3 (1907), pp. 389–398. MR 1500793 JFM 38.0180.02 article

It is here proved that, if \( p^{\pi} \) is the highest power of the prime \( p \) dividing the order a of a group \( G \), the group-matrix of \( G \) can be transformed, by a matrix whose elements are integers taken modulo \( p \), into a compound matrix in which the submatrices to the right of the main diagonal have zero elements throughout, while the \( p^{\pi} \) submatrices in the diagonal are identical. Let \( D \) denote one of the diagonal submatrices, so that \( D \) is a square matrix of order \( g/p^{\pi} \). Then the group-determinant \( \Delta \) of \( G \) is congruent to \( |D|^{p^{\pi}} \) modulo \( p \). This result is in marked contrast to the non-modular theory, in which each algebraically irreducible factor of \( \Delta \) enters to a power exactly equal to its degree.

It is shown in §8 that the group-matrices of all groups of order \( p^{\pi} \) can be reduced to their canonical form modulo \( p \) by one and the same transformation.

An interesting theorem on group-characters is obtained in §11.

[164]L. E. Dickson: “Invariants of the general quadratic form modulo \( {}2 \),” Proc. London Math. Soc. (2) 5 : 1 (1907), pp. 301–324. MR 1577339 JFM 38.0150.01 article

The theory of invariants of homogeneous forms under linear transformation with coefficients in a finite field of order \( p^n \) offers decided contrasts [1907] to the usual algebraic theory.

In the present paper I consider the invariants of the \( m \)-ary quadratic form [1907, 1908, 1899] \( Q_m \) in the field of order \( {}2 \), the most important case for the applications. For each value of \( m \) less than \( {}6 \), I obtain a complete set (\( m \) in number) of independent invariants, as well as a complete set of linearly independent invariants. A number of the invariants for \( m = 6 \) are given in §§18, 19.

The minimum number of variables upon which a form \( Q_m \) can be expressed equals the rank \( r \) of the discriminantal determinant, \( r \) being defined as in the algebraic theory, except that for minors of odd order the factor \( {}2 \) must first be deleted. When \( r \) is even there are two distinct canonical forms. It is shown in the present paper that the complete classification of quadratic forms can be accomplished by means of invariant functions.

[165]L. E. Dickson: “Errata: ‘Invariants of binary forms under modular transformations’,” Trans. Am. Math. Soc. 8 : 4 (1907), pp. 535. Errata for article in Trans. Am. Math. Soc. 8:2 (1907). MR 1500482 article

[166]L. E. Dickson: “Note on the volume of a tetrahedron in terms of the coordinates of the vertices,” Amer. Math. Mon. 14 : 6–7 (1907), pp. 117–118. MR 1516848 article

[167]L. E. Dickson: “Book reviews: Zahlentheorie and Einleitung in die allgemeine Theorie der Algebraischen Grössen,” Bull. Am. Math. Soc. 13 : 7 (1907), pp. 348–362. Books by P. Bachmann (Teubner, 1905) and J. König (Teubner, 1903). MR 1558477 article

Julius König	Related
Paul Bachmann	Related

[168]L. E. Dickson: “The symmetric group on eight letters and the senary first hypoabelian group,” Bull. Am. Math. Soc. 13 : 8 (1907), pp. 386–389. MR 1558486 JFM 38.0182.01 article

[169]L. E. Dickson: “On the last theorem of Fermat,” Messenger of Mathematics 38 (1908), pp. 14–32. Part II published in Quart. J. Pure Appl. Math. 40 (1908). JFM 39.0260.01 article

[170]L. E. Dickson: “On families of quadratic forms in a general field,” Quart. J. Pure Appl. Math. 39 (1908), pp. 316–333. JFM 39.0148.01 article

[171]L. E. Dickson: “The Galois group of a reciprocal quartic equation,” Amer. Math. Mon. 15 : 4 (April 1908), pp. 71–78. MR 1517010 JFM 39.0130.01 article

[172]L. E. Dickson: “On the congruence \( x^n+y^n+z^n=0 \) (mod \( p \)),” J. Reine Angew. Math. 135 (1908), pp. 134–141. JFM 39.0260.02 article

[173]L. E. Dickson: “On the last theorem of Fermat, II,” Quart. J. Pure Appl. Math. 40 (1908), pp. 27–45. Part I published in Messenger of Mathematics 38 (1908). JFM 39.0260.03 article

[174]L. E. Dickson: “On the canonical forms and automorphs of ternary cubic forms,” Amer. J. Math. 30 : 2 (April 1908), pp. 117–128. MR 1506034 JFM 39.0149.01 article

Gordan [1900] has given a complete set of canonical types of ternary cubic forms and has determined the algebraic irrationalities occurring in the reducing linear transformations. There does not seem to be at hand a reduction theory in which the coefficients of the form and those of the reducing transformations belong to a given field \( F \). The case in which \( F \) has the modulus \( {}3 \) is essentially different from the contrary case and will be treated in the present paper. After treating the reduction problem rationally in the initial field, we consider, in §§19–20, reductions involving irrationalities and obtain eleven ultimate canonical forms. The result for modular fields is in contrast to Gordan’s results for the field of all complex numbers.

[175]L. E. Dickson: “Invariantive reduction of quadratic forms in the \( \mathit{GF}[2^n] \),” Amer. J. Math. 30 : 3 (July 1908), pp. 263–281. MR 1506042 JFM 39.0146.01 article

In the American Journal of Mathematics, Vol. XXI (1899), I gave a complete set of non-equivalent canonical forms of \( m \)-ary quadratic forms in the Galois field of order \( p^n \). The cases \( p=2 \) and \( p > 2 \) are essentially different. In the opening pages of the present paper, I give a simpler treatment of the important case \( p=2 \), a treatment bringing to the front some of the invariants of the form. In §§4,5, I show that the rank \( r \) of the discriminantal determinant gives the minimum number of variables on which the form can be expressed. The definition of \( r \) in this modular theory differs from that in the algebraic theory in the employment of the halves of the minors of odd order. In particular, for \( m \) odd, the discriminant vanishes identically in the \( \mathit{GF}[2^n] \), while the semi-discriminant \( S_m \) is an important invariant.

The larger part of the paper is devoted to the determination and application of a complete set of linearly independent invariants of the ternary [1907] quadratic form \[ a_1x_2x_3 + \cdots + \sum b_ix_i^2 \] in the \( \mathit{GF}[2^n] \) for \( n\leq 4 \). All the invariants may be expressed in terms of three fundamental independent invariants: \begin{align*} S_3 &= a_1a_2a_3 + \sum a_i^2b_i,\\ A &= \prod_{i=1,2,3}(a_i^{2^n-1}-1),\\ F &= f + f^2 + f^4 + \cdots + f^{2^{n-1}}, \end{align*} where \( f \) is a function increasing rapidly in complexity as \( n \) increases.

[176]L. E. Dickson: “On the factorization of large numbers,” Amer. Math. Mon. 15 : 12 (December 1908), pp. 217–222. MR 1517123 JFM 39.0272.04 article

In the study of a difficult problem, it is a decided handicap to be denied the useful information that accompanies a knowledge of the origin of the proposed problem. There is little interest and much labor in the factorization of numbers taken at random. The real desideratum is a method which is capable of making effective use of the information which can be derived from the origin of the proposed number, and of auxiliary tables at command. For example, we may be concerned with numbers of a given form such as \( m^n\pm 1 \), or with the eliminant of a system of congruences under investigation. We shall here illustrate such a method by determining the composition, hitherto unknown, of two numbers each of eleven digits.

[177]L. E. Dickson: “Representations of the general symmetric group as linear groups in finite and infinite fields,” Trans. Am. Math. Soc. 9 : 2 (1908), pp. 121–148. MR 1500805 JFM 39.0198.02 article

In a series of articles in the Berliner Berichte, beginning in 1896, Frobenius has developed an elaborate theory of group-characters and applied it to the representation of a given finite group \( G \) as a non-modular linear group. Later, Burnside [1898, 1903] approached the subject from the standpoint of continuous groups. The writer has shown [1902] that the method employed by Burnside may be replaced by one involving only purely rational processes and hence leading to results valid for a general field. The last treatment, however, expressly excludes the case in which the field has a modulus which divides the order of \( G \). The exclusion of this case is not merely a matter of convenience, nor merely a limitation due to the particular method of treatment; indeed [1903], the properties of the group-determinant differ essentially from those holding when the modulus does not divide the order of \( G \). Thus when \( G \) is of order \( q! \), the general theory gives no information as to the representations in a field having a modulus \( {}\leq q \), whereas the case of a small modulus is the most important one for the applications.

The present paper investigates the linear homogeneous groups on \( m \) variables, with coefficients in a field \( F \), which are simply isomorphic with the symmetric group on \( q \) letters. The treatment is elementary and entirely independent of the papers cited above; in particular, the investigation is made for all moduli without exception. The principal result is the determination of the minimum value of the number of variables.

[178]L. E. Dickson: “Criteria for the irreducibility of a reciprocal equation,” Bull. Am. Math. Soc. 14 : 9 (1908), pp. 426–430. MR 1558646 JFM 39.0121.01 article

A reciprocal equation \( f(x) = x^m + \cdots = 0 \) is one for which \[ x^m f(1/x) \equiv cf(x). \] Replacing \( x \) by \( {}1/x \), we see that \( f \equiv c^2f \), \( c = \pm 1 \). Now \( f(x) \) has the factor \( x \pm 1 \) and hence is reducible, unless \( m \) is even and \( c = +1 \). Further discussion may therefore be limited to equations \[ F(x) \equiv x^{2n} + c_1 x^{2n-1} + c_2x^{2n-2} + \cdots + c_2x^2 + c_1x + 1 = 0 \] of even degree and having \[ x^{2n}F(1/x) \equiv F(x). \] Let \( R \) be a domain of rationality containing the \( c \)’s.

Under the substitution \[ x + 1/x = y, \] \( x^{-n}F(x) \) becomes a polynomial in \( y \), \[ \phi(y) = y^n + k_1y^{n-1} + \cdots + k_n, \] with coefficients in \( R \). By a suitable choice of the \( c \)’s, the \( k \)’s may be made equal to any assigned values.

We shall establish in §§2–7 the following:

Necessary and sufficient conditions for the irreducibility of \( F(x) \) in the domain \( R \) are

\( \phi(y) \) must be irreducible in \( R \).
\( F(x) \) must not equal a product of two distinct irreducible functions of degree \( n \).

The second condition is discussed in §§8–10.

[179]L. E. Dickson: “On triple algebras and ternary cubic forms,” Bull. Am. Math. Soc. 14 : 4 (1908), pp. 160–169. MR 1558578 JFM 39.0138.03 article

For any field \( F \) in which there is an irreducible cubic equation \( f(\rho) = 0 \), the norm of \( x + y\rho + z\rho^2 \) is a ternary cubic form \( C \) which vanishes for no set of values \( x, y, z \) in \( F \), other than \( x = y = z = 0 \). The conditions under which the general ternary form has the last property are here determined for the case of finite fields. One formulation of the result is as follows:

The necessary and sufficient conditions that a ternary cubic form \( C \) shall vanish for no set of values \( x \), \( y \), \( z \) in the \( \mathit{GF}[p^n] \), \( p > 2 \), other than \( x = y = z = 0 \), are that its Hessian shall equal \( mC \) where \( m \) is a constant different from zero, and that the binary form obtained from \( C \) by setting \( z = 0 \) shall be irreducible in the field.

[180]L. E. Dickson: “On higher congruences and modular invariants,” Bull. Am. Math. Soc. 14 : 7 (1908), pp. 313–318. MR 1558615 JFM 39.0256.04 article

[181]L. E. Dickson: “A class of groups in an arbitrary realm connected with the configuration of the \( {}27 \) lines on a cubic surface (second paper),” Quart. J. Pure Appl. Math. 39 (1908), pp. 205–209. JFM 39.0198.03 article

[182]L. E. Dickson: “Combinants,” Quart. J. Pure Appl. Math. 40 (1908), pp. 349–365. JFM 40.0164.01 article

[183]L. E. Dickson: “Rational edged cuboids with equal volumes and equal surfaces,” Amer. Math. Mon. 16 : 6–7 (June–July 1909), pp. 107–114. MR 1517231 article

[184]L. E. Dickson and M. Kaba: “On the representation of numbers as the sum of two squares,” Amer. Math. Mon. 16 : 5 (May 1909), pp. 85–87. JFM 40.0267.07 article

[185]L. E. Dickson: “On the representation of numbers by modular forms,” Bull. Am. Math. Soc. 15 : 7 (1909), pp. 338–347. MR 1558771 JFM 40.0269.01 article

For any field \( F \) in which there is an irreducible equation \( f(\rho) = 0 \) of degree \( m \), the norm of \[ x_0 + x_1\rho + x_2\rho^2 + \cdots + x_{m-1}\rho^{m-1} \] is a form of degree \( m \) in \( m \) variables which vanishes for no set of values \( x_i \) in the field \( F \), other than the set in which every \( x_i = 0 \). For a finite field it seems to be true that every form of degree \( m \) in \( m + 1 \) variables vanishes for values, not all zero, in the field. For \( m = 2 \) and \( m = 3 \) this theorem is established in §§2, 3. The corresponding theorem does not hold in general for infinite fields. But A. Meyer [Bachmann 1898, p. 266, p. 553] has shown that any indefinite quadratic form in five variables vanishes for integral values, not all zero, of the variables.

Modular forms which represent only squares have been investigated at length by the writer [1909]; those which represent cubes exclusively are considered in §§4–12.

[186]L. E. Dickson: “On commutative linear groups,” Quart. J. Pure Appl. Math. 40 (1909), pp. 167–196. JFM 40.0186.01 article

[187]L. E. Dickson: “Equivalence of pairs of bilinear or quadratic forms under rational transformation,” Trans. Am. Math. Soc. 10 : 3 (1909), pp. 347–360. MR 1500845 JFM 40.0163.01 article

[188]L. E. Dickson: “A theory of invariants,” Amer. J. Math. 31 : 4 (October 1909), pp. 337–354. MR 1506079 JFM 40.0157.01 article

Consider a system \( S \) of forms \( f_1,\dots, \) \( f_s \) on \( m \) variables, where \( f_i \) is the general polynomial of degree \( d_i \), having as coefficients arbitrary parameters in any given field \( F \), finite or infinite. Let \( L \) be any given group of \( m \)-ary linear homogeneous transformations with coefficients in \( F \). The particular systems \( S^{\prime} \), \( S^{\prime\prime},\dots, \) obtained from the general system \( S \) by assigning to the coefficients particular sets of values in the field, may be separated into classes \( C_i \) under the group \( L \), such that \( S^{\prime} \) and \( S^{\prime\prime} \) belong to the same class if and only if they are equivalent under \( L \).

We shall employ the term function in Dirichlet’s sense of correspondence and shall consider only single-valued functions taking exclusively values in the field \( F \).

Let the function \( \phi \) have one and only one value in the field for each of the systems \( S^{\prime} \), \( S^{\prime\prime},\dots \). In particular, if, for each \( i \), \( \phi \) has the same value \( v_i \) for all the systems in the class \( C_i \), then \( \phi \) is an invariant under the group \( L \).

For a finite field such that an invariant is a rational integral function of the coefficients of \( S \), an explicit expression for which is given in \( \S4 \). More convenient expressions may be given when \( S \) is a special system (§§10–18).

[189]L. E. Dickson: “Lower limit for the number of sets of solutions of \( x^e+y^e+z^e \equiv 0 \) (mod \( p \)),” J. Reine Angew. Math. 135 (1909), pp. 181–188. JFM 40.0254.04 article

[190]L. E. Dickson: “On certain diophantine equations,” Messenger of Mathematics 39 (1909), pp. 86–87. JFM 40.0254.03 article

[191]L. E. Dickson: “General theory of modular invariants,” Trans. Am. Math. Soc. 10 : 2 (1909), pp. 123–158. MR 1500831 JFM 40.0158.01 article

The discovery of the fundamental theorems, established in the first part of the present paper, on the invariants of a general system of \( s \) forms under linear transformations in a finite field was the outcome of a new standpoint for the consideration of modular invariants. In former papers on the subject (cited later), the test for the invariance of a polynomial consisted in a more or less direct verification that it remained unaltered, up to a power of the determinant of the transformation, under the general linear group \( G \) in the field; instead of certain generators of the latter, the corresponding annihilators were employed. In the present paper, the transformation concept is employed only to furnish a complete set of non-equivalent classes \( C_0,\dots, \) \( C_{f-1} \) of systems of \( s \) forms under the group \( G \). Thus the test for the absolute invariance of a polynomial \( P \) is that \( P \) shall take the same value for all systems of a forms in a class. It is shown in §4 that the number of linearly independent absolute invariants equals the number \( f \) of classes under the total group \( G \). In §6 it is shown that the number of linearly independent invariants, including both absolute and relative, equals the number of classes under the group \( G_1 \) of transformations of determinant unity; it is furthermore specified which of the invariants under \( G_1 \) are invariants of the \( s \) forms.

The general theory is applied in §§8, 9, 16–19 to the determination of all the invariants of the general \( m \)-ary quadratic form in the Galois field of order \( p^n \) and in §§22–26 to the construction of all invariants of the binary cubic form in the \( \mathit{GF}[p^n] \). For the practical construction of the invariants, there is developed a uniform process, of function-theoretic nature, for the conversion of non-invariantive characterizations of the classes into invariantive characterizations. The intervening sections are devoted to the determination and characterization of the classes of the forms under investigation. A mere list of canonical types of forms is not sufficient. For \( m \)-ary quadratic forms in the \( \mathit{GF}[2^n] \) such a list has been given by the author [1899, p. 222]; to obtain necessary and sufficient criteria for each class, a new theory for such forms has been constructed in §§10–15. Also for binary cubic forms, the case (§26) in which the modulus \( p \) equals \( {}2 \) is more intricate than the general case \( p > 2 \). The nature of the invariants is quite different in the two cases, a result to be anticipated for quadratic forms, but rather surprising for cubic forms. The consequent assignment of such a large part of the present paper to the special case \( p = 2 \) was made not merely for the sake of completeness, but rather on account of the very prominent rôle which the linear groups with modulus \( {}2 \) play in the applications [Jordan 1870, p. 313, p. 329] to geometry and in the general theory of linear groups.

[192]L. E. Dickson: “Definite forms in a finite field,” Trans. Am. Math. Soc. 10 : 1 (1909), pp. 109–122. MR 1500830 JFM 40.0268.03 article

[193]L. E. Dickson: “Rational reduction of a pair of binary quadratic forms; their modular invariants,” Amer. J. Math. 31 : 2 (April 1909), pp. 103–146. MR 1506065 JFM 40.0161.01 article

The primary object of the present paper is a study of the invariants of a pair of binary quadratic forms under modular transformation. Incidentally, the invariants of a single form are given a more satisfactory expression than hitherto employed (§7).

It is shown that the knowledge of a complete set of canonical types of pairs of forms is of great service in the discovery and proof of relations between certain of the modular invariants and in establishing the independence of other invariants (§§23,25). For these reasons and for the purpose of giving interpretations to the modular invariants, we begin the investigation with a discussion of the necessary and sufficient conditions for the equivalence of two pairs of quadratic forms.

[194]L. E. Dickson: “Book review: Diophantische Approximationen: Eine Einführung in die Zahlentheorie,” Bull. Am. Math. Soc. 15 : 5 (1909), pp. 251–252. Book by H. Minkowski (Teubner, 1907). MR 1558750 article

[195]L. E. Dickson: “Modular invariants of a general system of linear forms,” Proc. London Math. Soc. 7 : 1 (1909), pp. 430–444. MR 1575684 JFM 40.0160.01 article

[196]L. E. Dickson: “Book review: Manuscrits de Evariste Galois,” Bull. Am. Math. Soc. 15 : 5 (1909), pp. 249–250. Book by J. Tannery (Gauthier-Villars, 1908). MR 1558749 article

Evariste Galois	Related
Jules Tannery	Related

[197]L. E. Dickson: “Book review: The collected mathematical papers of James Joseph Sylvester (Vols. I and II),” Bull. Am. Math. Soc. 15 : 5 (1909), pp. 232–239. Books by H. F. Baker (Cambridge University Press, 1904 and 1908). A review of Volume III was published in Bull. Am. Math. Soc. 17:5 (1911). MR 1558745 article

Henry Frederick Baker	Related
James Joseph Sylvester	Related

[198]L. E. Dickson: “On the factorization of integral functions with \( p \)-adic coefficients,” Bull. Am. Math. Soc. 17 : 1 (1910), pp. 19–23. MR 1558975 JFM 41.0229.02 article

If \( F(X) \) is an integral function of degree \( n \) with integral \( p \)-adic coefficients, then for any integer \( k \) we have a congruence \begin{align*} F(x) &\equiv F^{(k)}(x) \pmod{p^{k+1}}\\ & = F_0(x) + pF_1(x) + p^2F_2(x) + \cdots +p^kF_k(x), \end{align*} in which each \( F_i(x) \) is an integral function of degree \( {}\leq n \) with coefficients belonging to the set \( {}0 \), \( {}1,\dots, \) \( p-1 \). The function \( F^{(k)}(x) \) is called the convergent of rank \( k \) of \( F(x) \). If \begin{equation*}\tag{1} F(x)-f(x).g(x) \mod{p} \end{equation*} in which the factors are integral functions with integral \( p \)-adic coefficients, then for any integer \( k \) we obviously have \[ F^{(k)}(x) \equiv f^{(k)}(x)\cdot g^{(k)}(x) \mod{p^{k+1}} \] The following converse theorem plays a fundamental rôle in Hensel’s new theory of algebraic numbers [1908, p. 71]: If \[ F(x) \equiv f_0(x)\cdot g_0(x) \mod{p^{s+1}} \] for \( s + 1 > 2\rho \), where \( \rho \) is the order of the resultant \( R \) of \( f_0(x) \) and \( g_0(x) \), then \( F(x) \) is the product (1) of two integral functions with integral \( p \)-adic coefficients whose convergents of rank \( s - \rho \) are \( f_0(x) \) and \( g_0(x) \).

Hensel’s proof is in effect a process to construct the successive convergents of \( f(x) \) and \( g(x) \). Each step of the process requires the solution of a linear equation in two unknowns with \( p \)-adic coefficients. The object of this note is to furnish a decidedly simpler process, which dispenses with these linear equations, and requires only the solution of a single linear congruence.

[199]L. E. Dickson: “Book review: Factor table for the first ten millions,” Bull. Am. Math. Soc. 17 : 1 (1910), pp. 36–38. Book by D. N. Lehmer (Carnegie Institution, 1909). MR 1558977 article

[200]L. E. Dickson: “Book review: Theorie der Algebraischen Zahlen,” Bull. Am. Math. Soc. 17 : 1 (1910), pp. 23–36. Book by K. Hensel (Teubner, 1908). MR 1558976 article

[201]L. E. Dickson: “On non-vanishing forms,” Quart. J. Pure Appl. Math. 42 (1911), pp. 162–171. JFM 42.0138.01 article

[202]L. E. Dickson: “Book review: Festschrift zur Feier des 100 Geburtstages Eduard Kummers mit Briefen an seine Mutter und an Leopold Kronecker,” Bull. Am. Math. Soc. 17 : 7 (1911), pp. 371–372. Book by K. Hensel (Teubner, 1910). MR 1559074 article

Leopold Kronecker	Related
Ernst Eduard Kummer	Related
Kurt Hensel	Related

[203]L. E. Dickson: “Note on cubic equations and congruences,” Ann. Math. (2) 12 : 3 (April 1911), pp. 149–152. MR 1503562 JFM 42.0214.02 article

[204]L. E. Dickson: “Notes on the theory of numbers,” Amer. Math. Mon. 18 : 5 (May 1911), pp. 109–111. MR 1517545 JFM 42.0236.12 article

[205]L. E. Dickson: “Note on modular invariants,” Quart. J. Pure Appl. Math. 42 (1911), pp. 158–161. JFM 42.0137.01 article

[206]L. E. Dickson: “An invariantive investigation of irreducible binary modular forms,” Trans. Am. Math. Soc. 12 : 1 (1911), pp. 1–18. MR 1500877 JFM 42.0134.02 article

A fundamental system of invariants of the group of all binary linear transformations in a finite field is shown in §§2–6 to consist of two invariants, one the product of the distinct linear forms and the other the product of the distinct irreducible quadratic forms, where in each case no two factors have a constant ratio. The product \( \pi_m \) of the irreducible forms of degree \( m \) can be expressed in terms of the fundamental invariants; this is accomplished in §§7–9 by means of the remarkable three-term recursion formula (15).

Two forms shall be said to belong to the same class if and only if one of them can be transformed into a constant multiple of the other by a linear transformation of determinant unity in the field. It is shown in §10 that there are as many classes of irreducible binary forms of degree \( m \) as there are irreducible factors of \( \pi_m \) when expressed as a function of a certain pair of invariants. The choice of the latter is different in the two cases, \( p = 2 \), \( p > 2 \), where \( p \) is the modulus of the field; this is due to the fact that, in the respective cases, there are one or two similarity transformations of determinant unity. The investigation is completed for the values of \( m \) less than \( {}8 \). The difficulties encountered increase as the number of factors of \( m \) increases. Certain problems arise for which the present invariantive theory affords an indirect solution, whereas a direct solution appears to be quite difficult (cf. end of §10, and end of §17). For \( m = 6 \), it was necessary to enumerate the irreducible cubics in the \( \mathit{GF}[p^n] \) whose roots are squares in the \( \mathit{GF}[p^{3n}] \) and have a given sum.

[207]L. E. Dickson: “A fundamental system of invariants of the general modular linear group with a solution of the form problem,” Trans. Am. Math. Soc. 12 : 1 (1911), pp. 75–98. MR 1500882 JFM 42.0136.01 article

We shall determine \( m \) functions which form a fundamental system of invariants for the group \( G_m \) of all linear homogeneous transformations on \( m \) variables with coefficients in the Galois field of order \( p^n \). In the so-called form problem for the group \( G_m \), we seek all sets of values of the \( m \) variables for which the \( m \) fundamental absolute invariants take assigned values. It is shown in §8 that all sets of solutions are linear combinations of the roots of an equation involving only the powers \( p^{nm} \), \( p^{n(m-1)},\dots, \) \( p^n \), \( {}1 \) of a single variable. This fundamental equation has properties analogous to those of a linear differential equation of the \( m \)-th order. In §§10–16 we determine the degrees of the irreducible factors of the fundamental equation and, in particular, the smallest field in which it is completely solvable. We obtain a wide generalization of the theory of the equation \( \xi^{p^{nm}}-\xi = 0 \), which forms the basis of the theory of finite fields. The function defined by the left member of the fundamental equation includes the type of substitution quantics in one variable the theory of which is equivalent to, but preceded historically, the theory of linear modular substitutions on m variables. We here find that the latter theory necessitates a return to the earlier quantics in one variable. Finally, in §§17–22, we consider the interpretation of certain invariants.

[208]L. E. Dickson: “Binary modular groups and their invariants,” Amer. J. Math. 33 : 1–4 (January 1911), pp. 175–192. MR 1507890 JFM 42.0157.03 article

In the first part of this paper I determine all subgroups of the group \( \Gamma \) composed of all binary transformations of determinant unity with coefficients in the Galois field \( F \) of order \( p^n \). The order of \( \Gamma \) is \[ \omega = p^n(p^{2n} - 1). \] I determined the subgroups of \( \Gamma \) in the spring of 1904 and made use of the results in investigating [1905a, 1906] the subgroups of the general ternary and quaternary linear groups modulo \( p \), as well as in my study of finite algebras [1905b].

The subgroups of \( \Gamma \) may be derived (as in §9) from the subgroups of the linear fractional group. We may however proceed independently (§§2–7). The latter method naturally brings out more clearly the properties of the homogeneous groups, and moreover furnishes material needed in the construction of the invariants (§§10–13). The linear fractional groups may be derived by inspection from the homogeneous groups.

The exceptional character of the case \( p=2 \) is more marked in the case of homogeneous groups than in the case of fractional groups. Moreover, the homogeneous and fractional groups are identical if \( p=2 \). For these reasons I assume here that \( p > 2 \).

[209]L. E. Dickson: “Book review: Diophantus of Alexandria,” Bull. Am. Math. Soc. 18 : 2 (1911), pp. 82–83. Book by T. L. Heath (Cambridge Univ. Press, 1910). MR 1559140 article

[210]L. E. Dickson: “Book review: Niedere Zahlentheorie, zweiter Teil,” Bull. Am. Math. Soc. 17 : 5 (1911), pp. 255–256. Book by P. Bachman (Teubner, 1910). MR 1559037 article

[211]L. E. Dickson: “Book review: The collected mathematical papers of James Joseph Sylvester (Vol. III),” Bull. Am. Math. Soc. 17 : 5 (1911), pp. 254–255. Book by H. F. Baker (Cambridge University Press, 1909). A review of Volumes I and II was published in Bull. Am. Math. Soc. 15:5 (1909). MR 1559036 article

Henry Frederick Baker	Related
James Joseph Sylvester	Related

[212]L. E. Dickson: “On the negative discriminants for which there is a single class of positive primitive binary quadratic forms,” Bull. Am. Math. Soc. 17 : 10 (1911), pp. 534–537. MR 1559109 JFM 42.0239.06 article

[213]L. E. Dickson: “Linear algebras,” Trans. Am. Math. Soc. 13 : 1 (1912), pp. 59–73. MR 1500905 JFM 43.0162.09 article

[214]L. E. Dickson: “Book review: Essai de géométrie analytique modulaire a deux dimensions,” Bull. Am. Math. Soc. 20 : 2 (1913), pp. 96–97. Book by G. Arnoux (Gauthier-Villars, 1911). MR 1559428 article

[215]L. E. Dickson: “On the rank of a symmetrical matrix,” Bull. Am. Math. Soc. 19 : 9 (1913), pp. 456. Abstract only. Abstract for article in Ann. Math. 15:1–4 (1913–1914). JFM 44.0177.03 article

[216]L. E. Dickson: “Even abundant numbers,” Amer. J. Math. 35 : 4 (October 1913), pp. 423–426. MR 1506195 JFM 44.0221.01 article

[217]L. E. Dickson: “Proof of the finiteness of modular covariants,” Trans. Am. Math. Soc. 14 : 3 (1913), pp. 299–310. An abstract was published in Bull. Am. Math. Soc. 19:9 (1913). MR 1500948 article

[218]L. E. Dickson: “Finiteness of the odd perfect and primitive abundant numbers with \( n \) distinct prime factors,” Bull. Amer. Math. Soc. 19 : 6 (1913), pp. 285. Abstract for article in Amer. J. Math. 35:4 (1913). JFM 44.0220.02 article

[219]L. E. Dickson: “On binary modular groups and their invariants,” Bull. Am. Math. Soc. 20 : 3 (1913), pp. 132–134. MR 1559435 JFM 44.0169.05 article

[220]L. E. Dickson: “Theorems and tables on the sum of the divisors of a number,” Quart. J. Pure Appl. Math. 44 (1913), pp. 264–296. JFM 44.0221.02 article

[221]L. E. Dickson: “Amicable number triples,” Amer. Math. Mon. 20 : 3 (1913), pp. 84–92. MR 1517797 JFM 44.0247.13 article

[222]L. E. Dickson: “The invariants, semivariants and linear covariants of the binary quartic form modulo \( {}2 \),” Ann. Math. (2) 15 : 1–4 (1913–1914), pp. 114–117. MR 1502467 JFM 45.0210.01 article

[223]L. E. Dickson: “Book review: Geometrie der Zahlen,” Bull. Am. Math. Soc. 21 : 3 (1914), pp. 131–132. Book by H. Minkowski (Teubner, 1910). MR 1559582 article

[224]L. E. Dickson: “Linear associative algebras and abelian equations,” Trans. Am. Math. Soc. 15 : 1 (1914), pp. 31–46. MR 1500963 JFM 45.0189.03 article

[225]L. E. Dickson: “On the trisection of an angle and the construction of regular polygons of \( {}7 \) and \( {}9 \) sides,” Amer. Math. Mon. 21 : 8 (1914), pp. 259–262. MR 1518094 JFM 45.0753.01 article

[226]L. E. Dickson: “Invariants in the theory of numbers,” Trans. Am. Math. Soc. 15 : 4 (1914), pp. 497–503. MR 1500992 JFM 45.0207.01 article

Polynomials in the coefficients of a form \( f(x_1,\dots, \) \( x_n) \) which have the invariantive property with respect to all linear homogeneous transformations on \( x_1,\dots, \) \( x_n \) with integral coefficients taken modulo \( p \), where \( p \) is a prime, are called formal invariants modulo \( p \) of \( f \) if the coefficients of \( f \) are independent variables, but are called modular invariants if the coefficients of \( f \) are integers taken modulo \( p \). The concept of formal invariants modulo \( p \) was introduced by Hurwitz [1903, p. 25]; but the only known results concerning them relate to the binary quadratic and cubic forms. On the contrary, a simple and effective theory of modular invariants has been given by the writer. A new method of deriving modular invariants from seminvariants is given in §8. But the main purpose of this paper is to present a simple general method of constructing formal invariants. The method is applicable also to formal seminvariants and, more generally, to the invariants of any linear congruence group (§7). Moreover, the new point of view forms an adequate basis for a general theory of formal invariants.

[227]L. E. Dickson: Algebraic invariants. Mathematical monographs 14. J. Wiley & Sons (New York), 1914. JFM 45.0196.10 book

[228]L. E. Dickson: Linear algebras. Cambridge Tracts in Mathematics and Mathematical Physics 16. Cambridge University Press, 1914. Republished in 1960. JFM 45.0189.01 book

[229]L. E. Dickson: Elementary theory of equations. J. Wiley & Sons (New York), 1914. JFM 45.0162.02 book

[230]L. E. Dickson: “Book review: Zahlentheorie,” Bull. Am. Math. Soc. 20 : 5 (1914), pp. 258–259. Book by K. Hensel (Göschen, 1913). MR 1559479 article

[231]L. E. Dickson: “Modular invariants of the system of a binary cubic, quadratic and linear form,” Quart. J. Pure Appl. Math. 45 (1914), pp. 373–384. JFM 45.1236.02 article

[232]L. E. Dickson: “The points of inflexion of a plane cubic curve,” Ann. Math. (2) 16 : 1–4 (1914–15), pp. 50–66. MR 1502488 JFM 45.0838.02 article

The object of the first half of this paper is to give a self-contained and elementary exposition of the geometrical side of the theory of the inflexion points of a cubic curve without singular points. The object of the second half is to present the algebraic side of the theory, including proofs that the equation \( X \) of the ninth degree to which the problem leads is solvable by radicals, a determination of the Galois group of \( X \) for certain special cubic curves and for the general one, and a proof that \( X \) can be solved by means of a quartic and two cubic equations and hence by means of three cube roots and four square roots, no one of which can be dispensed with in general.

This interesting problem affords an excellent illustration of the complete mastery over an intricate algebraic situation which is possible by the use of Galois’ theory of algebraic equations. Readers having little or no aquaintance with that theory will be able to see from this illuminating concrete example what the theory really means and what it can accomplish.

While the earlier results in the present paper are classic, the methods employed are largely novel and the exposition is expecially elementary. The material in §§11–15 relating to the definitive determination of the Galois group is believed to be new.

[233]L. E. Dickson: “Projective classification of cubic surfaces modulo \( {}2 \),” Ann. Math. (2) 16 : 1–4 (1914–1915), pp. 139–157. MR 1502501 JFM 45.0212.01 article

[234]L. E. Dickson: “Invariants, seminvariants, and covariants of the ternary and quaternary quadratic form modulo \( {}2 \),” Bull. Am. Math. Soc. 21 : 4 (1915), pp. 174–179. MR 1559607 JFM 45.0208.01 article

A simple and complete theory of seminvariants of a binary form modulo \( p \) was given in the writer’s second lecture at the Madison Colloquium [1914]. A fundamental system of covariants of a ternary quadratic form \( F \) modulo \( {}2 \) was obtained in the fourth lecture. In place of the method employed there (pages 77–79) to obtain the leading coefficient of a covariant of \( F \), we shall now present a simpler method which makes it practicable to treat also the corresponding question for quaternary quadratic forms. The new method is, moreover, in closer accord with the underlying principle of those lectures, viz., to place the burden of the determination of the modular invariants upon the separation of the ground forms into classes of forms equivalent under linear transformation. By making the utmost use of this principle, we shall obtain a simpler solution of the problem for the ternary case and then treat the new quaternary case.

[235]L. E. Dickson: “The straight lines on modular cubic surfaces,” Proc. Nat. Acad. Sci. U.S.A. 1 : 4 (April 1915), pp. 248–253. JFM 45.0212.02 article

In ordinary space a cubic surface without singular points contains exactly \( {}27 \) straight lines, of which \( {}27 \), \( {}15 \), \( {}7 \), or \( {}3 \) are real; there are \( {}45 \) sets of three coplanar lines, the three of no set being concurrent. In modular space, in which the coordinates of points and the coefficients of the equations of lines or surfaces are integers or Galois imaginaries taken modulo \( {}2 \), it is interesting to notice that three coplanar lines on a cubic surface may be concurrent (§2). A point with integral coordinates is called real. A line or surface is called real if the coefficients of its equations are integers. In space with modulus \( {}2 \), the number of real straight lines on a cubic surface without singular points is \( {}15 \), \( {}9 \), \( {}5 \), \( {}3 \), \( {}2 \), \( {}1 \), or \( {}0 \).

We shall give here an elementary, self-contained, investigation of some of the most interesting cubic surfaces modulo \( {}2 \). A complete classification of all such surfaces under real linear transformation will appear in the Annals of Mathematics, but without the present investigation of the configuration of their lines.

[236]L. E. Dickson: “Invariantive classification of pairs of conics modulo \( {}2 \),” Amer. J. Math. 37 : 4 (October 1915), pp. 355–358. An abstract was published as Bull. Am. Math. Soc. 22:1 (1915). MR 1506263 JFM 45.0210.02 article

With a conic \( F \) modulo \( {}2 \) is associated covariantively a point \( A \), called its apex, and a unique line \( L \), and conversely \( A \) and \( L \) uniquely determine \( F \) (Madison Colloquium Lectures, 1914, page 69). Hence the projective classification of pairs of conics \( F \) and \( F^{\prime} \) is equivalent to that of the systems \( A \), \( L \), \( A^{\prime} \), \( L^{\prime} \) of two points and two lines and the degenerate systems in which one or more of the four elements are absent. A simple geometrical discussion of such systems leads to the theorem: Two pairs of conics modulo \( {}2 \) are protectively equivalent if and only if they have the same properties as regards existence of apices and covariant lines, distinctness of apices and lines, and incidence of apices and lines. These properties are expressed analytically by very simple modular invariants, which therefore form a fundamental system of modular invariants of two conics.

[237]L. E. Dickson: “Quartic curves modulo \( {}2 \),” Trans. Am. Math. Soc. 16 : 2 (April 1915), pp. 111–120. MR 1501003 JFM 45.0211.02 article

Let \( f(x,y,z) \) be a homogeneous form of order \( n \) with integral coefficients. The ponts for which the three partial derivatives of \( f \) are congruent to zero modulo \( {}2 \) shall be called derived points. A derived point shall be called a singular point or an apex of \( f=0 \) according as it is or is not on \( f=0 \). Apices do not arise if \( n \) is odd, since the left member of Euler’s relation \[ x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} + z\frac{\partial f}{\partial z} = nf \] is zero at a derived point and therefore also \( f \) is zero. But if \( n \) is even, a derived point may not be on \( f=0 \) and thus be an apex.

For example, any non-degenerate conic modulo \( {}2 \) can be transformed linearly into \( x^2 + yz = 0 \). Its single derived point (\( {}100 \)) is an apex.

Quartic curves modulo \( {}2 \) have the remarkable property of possessing at most seven bitangents (or an infinity in a special case), whereas an algebraic quartic curve possesses twenty-eight in general. For the special quartic \( \beta \) of §4, any line through the apex (\( {}001 \)) is a bitangent, just as any line through the apex of the conic \( x^2 + yz = 0 \) is a tangent.

The number of non-equivalent types of quartic curves containing \( {}0 \), \( {}7 \), \( {}6 \) real points and having no real linear factor is \( {}8 \), \( {}1 \), \( {}6 \), respectively. In each case, the types are completely distinguished by the number and reality of the singular points and apices. Except for two types, in which there are only two bitangents and only two derived points, the intersections of the bitangents coincide completely with the derived points. The problem is more complicated in the case of quartic curves with five real points, there being twenty-five types (§7). Quartics with \( {}1 \), \( {}2 \), \( {}3 \), or \( {}4 \) real points have not been treated since they would probably not present sufficient novelty to compensate for the increased length of the investigation.

[238]L. E. Dickson: “A tribute to Mildred Lenora Sanderson,” Amer. Math. Mon. 22 : 8 (1915), pp. 264. MR 1518319 article

[239]L. E. Dickson: “On the relation between linear algebras and continuous groups,” Bull. Am. Math. Soc. 22 : 2 (1915), pp. 53–61. MR 1559711 JFM 45.0189.02 article

[240]L. E. Dickson: “Invariantive theory of plane cubic curves modulo \( {}2 \),” Amer. J. Math. 37 : 2 (April 1915), pp. 107–116. MR 1506248 JFM 45.0210.03 article

The ten types of plane cubic curves in ordinary geometry have been characterized by invariants and covariants by Gordan [1900]. The types in modular geometry can be characterized by invariants only, the abundance of invariants making it unnecessary to resort to covariants. The most effective theory of modular invariants is that based upon a separation of the particular cases of the form in question into classes of equivalent forms.

For the present problem of cubic curves modulo \( {}2 \), this classification is effected in §3 by means of the real points (i.e., points with integral coördinates) on the cubic, supplemented by a determination of the real inflexion points and the real and imaginary singular points. While we could test directly each real point on the curve, not a singular point, and find whether or not it is an inflexion point, we have completed the geometrical investigation by making a determination of all of the real and imaginary inflexion points on each of the twenty-two types of cubic curves modulo \( {}2 \). For this purpose we have set up in §2 a cubic function \( H \), which here plays a rôle analogous to that played by the Hessian in the algebraic theory.

From the geometrical classification of the modular cubics we easily derive in §4 a fundamental system of modular invariants.

The methods employed in this paper are applicable to other problems of this nature; they indicate the decided advantage to be gained in the theory of modular invariants from modular geometry as developed by Bussey and Veblen, Coble and the writer.

[241]L. E. Dickson: “Book review: List of prime numbers from \( {}1 \) to \( {}10{,}006{,}721 \),” Bull. Am. Math. Soc. 21 : 7 (1915), pp. 355–356. Book by D. N. Lehmer (Carnegie Institution, 1914). MR 1559652 article

[242]L. E. Dickson: “Geometrical and invariantive theory of quartic curves modulo \( {}2 \),” Amer. J. Math. 37 : 4 (October 1915), pp. 337–354. MR 1507897 JFM 45.0211.01 article

[243]L. E. Dickson: “Classification of quartic curves, modulo \( {}2 \),” Messenger of Mathematics 44 (1915), pp. 189–192. JFM 45.1235.03 article

[244]L. E. Dickson: “Recent progress in the theories of modular and formal invariants and in modular geometry,” Proc. Nat. Acad. Sci. U.S.A. 1 : 1 (January 1915), pp. 1–4. JFM 45.0208.02 article

[245]L. E. Dickson: “Invariantive classification of pairs of conics modulo \( {}2 \),” Bull. Amer. Math. Soc. 22 : 1 (1915), pp. 4. Abstract only. Abstract for article in Amer. J. Math. 37:4 (1915). JFM 45.1361.01 article

With a conic \( F \) modulo \( {}2 \) is associated covariantively a point \( A \), called its apex, and a unique line \( L \), and conversely \( A \) and \( L \) uniquely determine \( F \) (Madison Colloquium Lectures, 1914, page 69). Hence the projective classification of pairs of conics \( F \) and \( F^{\prime} \) is equivalent to that of the systems \( A \), \( L \), \( A^{\prime} \), \( L^{\prime} \) of two points and two lines and the degenerate systems in which one or more of the four elements are absent. A simple geometrical discussion of such systems leads to the theorem: Two pairs of conics modulo \( {}2 \) are protectively equivalent if and only if they have the same properties as regards existence of apices and covariant lines, distinctness of apices and lines, and incidence of apices and lines. These properties are expressed analytically by very simple modular invariants, which therefore form a fundamental system of modular invariants of two conics.

[246]L. E. Dickson: “Book reviews: The theory of numbers and Diophantine analysis,” Bull. Am. Math. Soc. 22 : 6 (1916), pp. 303–310. Books by R. D. Carmichael (Wiley, 1914 and 1915). MR 1559782 article

[247]L. E. Dickson: “An extension of the theory of numbers by means of correspondences between fields,” Bull. Am. Math. Soc. 23 : 3 (1916), pp. 109–111. MR 1559876 JFM 46.0171.01 article

[248]G. A. Miller, H. F. Blichfeldt, and L. E. Dickson: Theory and applications of finite groups. J. Wiley & Sons (New York), 1916. Republished in 1938 and in 1961. JFM 46.0171.02 book

Hans Frederik Blichfeldt	Related
George A. Miller	Related

[249]L. E. Dickson: “Fermat’s last theorem and the origin and nature of the theory of algebraic numbers,” Ann. Math. (2) 18 : 4 (1917), pp. 161–187. MR 1503597 JFM 46.0268.02 article

[250]L. E. Dickson: “Book review: Vom periodischen Dezimalbruch zur Zahlentheorie,” Bull. Am. Math. Soc. 23 : 7 (1917), pp. 324–325. Book by A. Leman (Teubner, 1916). MR 1559951 article

[251]L. E. Dickson: History of the theory of numbers, vol. I: Divisibility and primality. Carnegie Institution (Washington, DC), 1919. See also Volume II and Volume III. Chelsea republished in 1966. The whole series was republished in 1934. JFM 47.0100.04 book

[252]L. E. Dickson: “On quaternions and their generalization and the history of the eight square theorem. Addenda.,” Ann. Math. (2) 20 : 4 (July 1919), pp. 297. Addenda to article in Ann. Math. 20:3 (1919). MR 1502566 article

[253]L. E. Dickson: “On quaternions and their generalization and the history of the eight square theorem,” Ann. Math. (2) 20 : 3 (1919), pp. 155–171. Addenda published in Ann. Math. 20:4 (1919). MR 1502549 article

We shall present the history of the generalization to four and eight squares of the familiar fomula \begin{align*} (a^2+b^2)(\alpha^2+\beta^2) &= r^2+s^2,\\ r &= a\alpha - b\beta,\\ s &= a\beta + b\alpha, \end{align*} and an elementary exposition of Hurwitz’s proof that such a formula holds only for \( {}2 \), \( {}4 \) or \( {}8 \) squares. For these three cases we shall show that the formula admits of a simple interpretation concerning the norms of numbers which are ordinary complex numbers, quaternions or numbers of Cayley’s algebra with \( {}8 \) units. No knowledge of quaternions or the latter algebra will be presupposed, but their more fundamental algebraic properties will be developed in detail.

A clear exposition will be given (§§1–5) of the main results of our subject. This will be followed (§§6–28) by an account of its history, which is believed to omit no paper on the eight square theorem and its generalization.

[254]L. E. Dickson: “Mathematics in war perspective: Presidential address delivered before the American Mathematical Society, December 27, 1918,” Bull. Am. Math. Soc. 25 : 7 (1919), pp. 289–311. MR 1560190 JFM 47.0046.06 article

[255]L. E. Dickson: “Applications of the geometry of numbers to algebraic numbers,” Bull. Am. Math. Soc. 25 : 10 (1919), pp. 453–455. MR 1560222 JFM 47.0894.01 article

[256]L. E. Dickson: “Les polynomes égaux à des déterminants,” C. R. Acad. Sci. Paris 171 (1920), pp. 1360–1362. JFM 47.0080.04 article

[257]L. E. Dickson: History of the theory of numbers, vol. II: Diophantine analysis. Carnegie Institution (Washington, DC), 1920. See also Volume I and \xlink{Volume III|}. Chelsea republished in 1966. The whole series was republished in 1934. book

[258]L. E. Dickson: “Fallacies and misconceptions in Diophantine analysis,” Bull. Am. Math. Soc. 27 : 7 (1921), pp. 312–319. MR 1560425 JFM 48.0137.03 article

[259]L. E. Dickson: “Reducible cubic forms expressible rationally as determinants,” Ann. Math. (2) 23 : 1 (1921), pp. 70–74. MR 1502596 JFM 48.0099.04 article

Let \( q \) be a quadratic form in four variables with rational coefficients.

If \( q \) vanishes at some rational point having \( y=0 \), \( yq \) is expressible rationally in determinantal form.
If \( q\neq 0 \) for every rational point having \( y\neq 0 \), then \( yq \) is expressible rationally in determinantal form if and only if either \( yq \) is equivalent to a ternary form, or the determinant of \( q \) is the square of a rational number \( \neq 0 \) and the determinant of \( q(x,0,z,w) \) is \( \neq 0 \).
If both of the preceding hypotheses be denied, so that \( q\neq 0 \) at every rational point having \( y=0 \), and \( q=0 \) for some rational point having \( y\neq 0 \), then \( yq \) is not expressible rationally in determinantal form.

[260]L. E. Dickson: “Arithmetic of quaternions,” Bull. Am. Math. Soc. 27 : 7 (1921), pp. 300. Abstract only. Abstract for article published in Proc. London Math. Soc. 20:1 (1922). JFM 48.0130.06 article

A. Hurwitz (Göttinger Nachrichten, 1896, p. 313) proved that the laws of arithmetic hold for integral quaternions, viz. those whose coordinates are either all integers or all halves of odd integers. Since fractions introduce an inconvenience in applications to Diophantine analysis, it is here proposed to define an integral quaternion to be one whose coordinates are all integers. It is called odd if its norm is odd. It is proved that, if at least one of two integral quaternions \( a \) and \( b \) is odd, they have a right-hand greatest common divisor \( d \) which is uniquely determined up to a unit factor (\( \pm 1 \), \( \pm i \), \( \pm j \), \( \pm k \)), and that integral quaternions \( A \) and \( B \) can be found such that \( d = Aa + Bb \). Similarly there is a left-hand greatest common divisor expressible in the form \( a\alpha + b\beta \). The further theory proceeds essentially as in Hurwitz’s exposition.

[261]L. E. Dickson: “Homogeneous polynomials with a multiplication theorem,” pp. 215–230 in Comptes rendus du Congrès International des Mathématiciens, 1920 (Strasbourg, 22–30 September 1920). Edited by H. Villat. Édouard Privat (Toulouse), 1921. Available open access here. JFM 48.1134.09 incollection

[262]L. E. Dickson: “Quaternions and their generalizations,” Proc. Nat. Acad. Sci. U.S.A. 7 : 4 (1921), pp. 109–114. JFM 48.0125.02 article

The discovery of quaternions by W. R. Hamilton in 1843 has led to an extensive theory of linear algebras (or closed systems of hypercomplex numbers) in which the quaternion algebra plays an important rôle. Frobenius [1878, p. 59] proved that the only real linear associative algebras in which a product is zero only when one factor is zero are the real number system, the ordinary complex number system, and the algebra of real quaternions. A much simpler proof has been given by the writer [1914]. Later, the writer [1915] was led to quaternions very naturally by means of the four parameter continuous group which leaves unaltered each line of a set of rulings on the quadric surface \[ x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 .\]

The object of the present note is to derive the algebra of quaternions and its direct generalizations without assuming the associative or commutative law. I shall obtain this interesting result by two distinct methods.

[263]L. E. Dickson: “La composition des polynomes,” C. R. Acad. Sci. Paris 172 (1921), pp. 636–640. JFM 48.0078.03 article

[264]L. E. Dickson: “Determination of all general homogeneous polynomials expressible as determinants with linear elements,” Trans. Am. Math. Soc. 22 : 2 (1921), pp. 167–179. MR 1501168 JFM 48.0099.02 article

The general quadratic forms in three and four variables can be transformed into \( x_1x_2-x_3^2 \) and \( x_1x_2-x_3x_4 \), respectively, and hence are expressible as determinants of order \( {}2 \). Since any binary form of degree \( r \) is a product of \( r \) linear forms, it is expressible as an \( r \)-rowed determinant whose elements outside the main diagonal are all zero.

It was proved geometrically by H. Schröter [1863] and more simply by L. Cremona [1868] that a sufficiently general cubic surface \( f = 0 \) is the locus of the intersections of corresponding planes of three projective bundles of planes: \[ \kappa l_{i1} + \lambda l_{i2} + \mu l_{i3} = 0 \qquad (i=1,2,3),\] where \( \kappa \), \( \lambda \), \( \mu \) are parameters and the \( l_{ij} \) are linear homogeneous functions of \( x_1,\dots, \) \( x_4 \). Hence \( f = 0 \) has the determinantal form \( |l_{ij}| = 0 \). Taking \( x_4 = 0 \), we see that a general cubic curve is expressible in determinantal form.

I shall prove that every plane curve is expressible in determinantal form and that, aside from the cases mentioned above, no further general homogeneous polynomial is expressible in determinantal form.

[265]L. E. Dickson: “Rational triangles and quadrilaterals,” Amer. Math. Mon. 28 : 6–7 (June–July 1921), pp. 244–250. MR 1519790 JFM 48.0667.04 article

[266]L. E. Dickson: “Some relations between the theory of numbers and other branches of mathematics,” pp. 41–56 in Comptes rendus du Congrès International des Mathématiciens, 1920 (Strasbourg, 22–30 September, 1920). Edited by H. Villat. Édouard Privat (Toulouse), 1921. JFM 48.1151.01 incollection

[267]L. E. Dickson: “A new method in Diophantine analysis,” Bull. Am. Math. Soc. 27 : 8 (1921), pp. 353–365. MR 1560438 JFM 48.0137.04 article

In the preceding number of this Bulletin (p. 312) I gave reasons why due caution should be observed toward the literature on the solution of homogeneous equations in integers. The valid knowledge concerning this subject is much less than has been usually admitted. The lack of general methods is even greater than in the subject of non-homogeneous equations. The chief aim of the present paper is to suggest such a method, based on the theory of ideals. The method is applicable in simple cases (§§2–4) without introducing ideals.

For the sake of brevity we shall restrict attention to the problem of finding all integral solutions of the equation \[ x_1^2 + ax_2^2 + bx_3^2 = x_4^2, \] an equation admitted [Carmichael 1915, p. 38] to be difficult of treatment by any known methods, and previously solved completely in integers only in the single case \( a = b = 1 \).

[268]L. E. Dickson: “A fundamental system of covariants of the ternary cubic form,” Ann. Math. (2) 23 : 1 (Sepbember 1921), pp. 78–82. MR 1502598 JFM 48.0099.03 article

In many different mathematical investigations use is made of covariants of the ternary cubic form \( F \). Less frequent use is made of the further concomitants involving line coördinates, and these will not be discussed here. The complete system of the \( {}34 \) concomitants was obtained by symbolic methods by Clebsch and Gordan [1873] and simpler by Gundelfinger [1871]. They were exhibited in non-symbolic form by Cayley [1881] for the canonical form \[ \sum a_i x_i^3 + 6lx_1x_2x_3 .\] Certain concomitants are obtained in the texts by Salmon, Elliott, and Weber, but no attempt is made to find a fundamental system.

The object of the present paper is to prove by an elementary method that fundamental system of covariants of \( F \) is given by \( F \), two invariants \( S \) and \( T \), the Hessian \( H \) of \( F \), the bordered Hessian determinant \( G \), and the Jacobian \( J \) of \( F \), \( H \), \( G \): \begin{gather*} 6^3H = \begin{vmatrix} F_{11} & F_{12} & F_{13} \\ F_{21} & F_{22} & F_{23} \\ F_{31} & F_{32} & F_{33} \end{vmatrix} \\ 6^2G = \begin{vmatrix} F_{11} & F_{12} & F_{13} & H_1 \\ F_{21} & F_{22} & F_{23} & H_2 \\ F_{31} & F_{32} & F_{33} & H_3 \\ H_1 & H_2 & H_3 & 0 \end{vmatrix} \\ 9J = \begin{vmatrix} F_1 & H_1 & G_1 \\ F_2 & H_2 & G_2 \\ F_3 & H_3 & G_3 \end{vmatrix} \end{gather*} where \( F_{ij} \) denotes \( \partial^2 F/\partial x_i\partial x_j \) and \( H_i \) denotes \( \partial H/\partial x_i \). The method enables us to compute anew the expressions for \( S \) and \( T \), and to deduce the syzygy between them and the covariants.

[269]L. E. Dickson: “Algebraic theory of the expressibility of cubic forms as determinants, with application to Diophantine analysis,” Amer. J. Math. 43 : 2 (April 1921), pp. 102–125. MR 1506433 JFM 48.0100.01 article

It was proved geometrically by H. Schröter [1863] and more simply by L. Cremona [1868, p. 79] that a sufficiently general cubic surface \( f = 0 \) is the locus of the intersections of corresponding planes of three projective bundles of planes: \begin{align*} \kappa l_{11} + \lambda l_{12} + \mu l_{13} &= 0,\\ \kappa l_{21} + \lambda l_{22} + \mu l_{23} &= 0,\\ \kappa l_{31} + \lambda l_{32} + \mu l_{33} &= 0, \end{align*} where \( \kappa \), \( \lambda \), \( \mu \) are parameters and the \( l_{ij} \) are linear homogeneous functions of \( x_1,\dots, \) \( x_4 \). Hence the surface is expressible in determinantal form \( |l_{ij}| = 0 \).

With the application to Diophantine analysis in mind, I here discuss algebraically the problem to express a given cubic form \( f \) as a determinant \( |l_{ij}| \) of the third order.

[270]L. E. Dickson: A first course in the theory of equations. J. Wiley & Sons (New York), 1922. Republished in 1939. JFM 48.0091.05 book

[271]L. E. Dickson: “Impossibility of restoring unique factorization in a hypercomplex arithmetic,” Bull. Am. Math. Soc. 28 : 9 (1922), pp. 438–442. MR 1560614 JFM 48.0125.03 article

[272]L. E. Dickson: Plane trigonometry with practical applications. B. H. Sanborn & Co. (Chicago), 1922. Republished in 1970. JFM 48.0672.13 book

[273]L. E. Dickson: “Integral solutions of \( x^2-my^2=zw \),” Bull. Am. Math. Soc. 29 : 10 (1923), pp. 464–467. MR 1560793 JFM 49.0096.04 article

[274]L. E. Dickson: “Should book reviews be censored?,” Amer. Math. Mon. 30 : 5 (1923), pp. 252–255. MR 1520244 article

[275]L. E. Dickson: “The rational linear algebras of maximum and minimum ranks,” Bull. Am. Math. Soc. 29 : 3 (1923), pp. 120–121. Abstract only. Abstract for article published in Proc. London Math. Soc. 22:1 (1923). Available open access here. JFM 49.0088.03 article

The investigation relates to linear associate algebras with a principal unit, the coordinates of whose numbers range over an arbitrary field \( F \). Such an algebra in \( n \) units of rank \( n \) is irreducible with respect to \( F \) if and only if \( f(x) \) is a power of a polynomial irreducible in \( F \), where \( f(x) = 0 \) is the rank equation. An algebra over \( F \), of rank \( {}2 \), has units \( {}1 \), \( e_i,\dots, \) \( e_m \) such that \begin{align*} e_i^2 = c_i,\\ e_ie_j = -e_je_i \quad(i\neq j), \end{align*} where \( c_i \) belongs to \( F \). If at least two \( c_i \) are not zero, then \( m = 3 \) and the algebra is the rational generalization of quaternions. If \( c_i = 0 \) (\( i < m \)), \( c_m\neq 0 \), the algebra is equivalent either to \begin{align*} & e_m^2 = 1,\\ & e_m e_i = e_i &&(i\leq k),\\ & e_m e_j = -e_j &&(k < j < m),\\ & e_r e_s = 0 &&(r < m, s < m), \end{align*} where \( {}2k \geq m - 1 \), or to \begin{align*} & e_m e_{2i-1} = e_{2i},\\ & e_m e_{2i} = c_me_{2i-1} &&(i\leq r),\\ & e_j e_k = 0 &&(j < m, k < m), \end{align*} with \( m = 2r + 1 \), and \( c_m \) not a square in \( F \). Finally, if every \( c_i = 0 \), the square of every number of the algebra is zero (a case of the outstanding problem of nilpotent algebras), and the rather numerous algebras in fewer than eight units are found. These results are applied to the determination of all algebras over any field \( F \) in \( {}2 \), \( {}3 \) or \( {}4 \) units.

[276]L. E. Dickson, H. H. Mitchell, H. S. Vandiver, and G. E. Wahlin: Algebraic numbers: Report of the committee on algebraic numbers. Bulletin of the National Research Council 28. National Research Council (Washington DC), 1923. Republished (in expanded form) as Algebraic numbers (1967). JFM 49.0109.01 book

Harry Schultz Vandiver	Related
George Eric Wahlin	Related
Howard Hawks Mitchell	Related

[277]L. E. Dickson: “A new simple theory of hypercomplex integers,” J. Math. Pure Appl. 2 : 9 (1923), pp. 281–326. An abstract was published in Bull. Am. Math. Soc. 29:3 (1923). JFM 49.0089.01 article

The definition of a system of hypercomplex integers due to A. Hurwitz and applied to all classic algebras in \( {}2 \), \( {}3 \) and \( {}4 \) units by Du Pasquier postulates rational coordinates, a finite arithmetical basis, closure under addition, subtraction and multiplication, the presence of the \( n \) basal units \( e_0 = 1,\dots, \) \( e_n \) (or only of \( e_0 \)), and that the system is a maximal. Unfortunately, no maximal system exists for the majority of algebras. If we employ any non-maximal system, it usually happens that factorization into primes is not unique and cannot be made unique by the introduction of ideals of any kind. These essential difficulties all disappear if we replace the postulate of a finite basis by the assumption that, for every number of the system of integers, the coefficients of the rank equation are all rational integers.

[278]L. E. Dickson: Algebras and their arithmetics. University of Chicago Press, 1923. Republished in 1938 and 1960. German translation published as Algebren und ihre Zahlentheorie (1927). JFM 49.0079.01 book

[279]L. E. Dickson: History of the theory of numbers, vol. III: Quadratic and higher forms. Carnegie Institution (Washington, DC), 1923. With a chapter on the class number by G. H. Cresse. See also Volume I and Volume II. Chelsea reprinted in 1966. The whole series was republished in 1934. JFM 49.0100.12 book

[280]L. E. Dickson: “A new simple theory of hypercomplex integers,” Bull. Am. Math. Soc. 29 : 3 (1923), pp. 121. Abstract of article published in J. Math. Pure Appl. 2:9 (1923). Available open access here. article

The definition of a system of hypercomplex integers due to A. Hurwitz and applied to all classic algebras in \( {}2 \), \( {}3 \) and \( {}4 \) units by Du Pasquier postulates rational coordinates, a finite arithmetical basis, closure under addition, subtraction and multiplication, the presence of the \( n \) basal units \( e_0 = 1,\dots, \) \( e_n \) (or only of \( e_0 \)), and that the system is a maximal. Unfortunately, no maximal system exists for the majority of algebras. If we employ any non-maximal system, it usually happens that factorization into primes is not unique and cannot be made unique by the introduction of ideals of any kind. These essential difficulties all disappear if we replace the postulate of a finite basis by the assumption that, for every number of the system of integers, the coefficients of the rank equation are all rational integers.

[281]L. E. Dickson: “Algebras and their arithmetics,” Bull. Am. Math. Soc. 30 : 5–6 (1924), pp. 247–257. MR 1560885 JFM 50.0631.02 article

[282]L. E. Dickson: “Quadratic fields in which factorization is always unique,” Bull. Am. Math. Soc. 30 : 7 (1924), pp. 328–334. MR 1560910 JFM 50.0113.03 article

[283]L. E. Dickson: “On the theory of numbers and generalized quaternions,” Amer. J. Math. 46 : 1 (1924), pp. 1–16. An abstract was published in Bull. Am. Math. Soc. 30:5 (1924). MR 1506514 JFM 50.0094.02 article

[284]L. E. Dickson: “Book review: Vorlesungen über die Theorie der Algebraischen Zahlen,” Amer. Math. Mon. 31 : 1 (January 1924), pp. 45–46. Book by E. Hecke (Akad. Verlagsges, 1923). MR 1520346 article

[285]L. E. Dickson: “Differential equations from the group standpoint,” Ann. Math. (2) 25 : 4 (1924), pp. 287–378. MR 1502670 article

The various classic devices for the integration of differential equations may be explained simply from a single standpoint — that of infinitesimal transformations leaving the equations invariant. What is still more important than this unification of diverse known methods, infinitesimal transformations furnish us a new tool, likely to succeed when the ordinary methods fail, since they enable us to take into account vital information ignored by the ordinary methods.

Although no previous acquaintance with differential equations is presupposed, the paper is not proposed as a substitute for the usual introductory course, but rather to provide a satisfactory review ab initio and at the same time to present the unifying and effective method based on groups.

[286]L. E. Dickson: “Book review: Die Arithmetik der quadratischen Formen, zweite Abteilung,” Bull. Am. Math. Soc. 30 : 7 (1924), pp. 373. Book by P. Bachmann (Teubner, 1923). MR 1560922 article

[287]L. E. Dickson: “On the theory of numbers and generalized quaternions,” Bull. Amer. Math. Soc. 30 : 5 (1924), pp. 228. Abstract only. Abstract for Amer. J. Math. 46:1. JFM 50.0095.08 article

[288]L. E. Dickson: “Book review: History of mathematics,” Amer. Math. Mon. 32 : 10 (December 1925), pp. 511–512. Book by D. E. Smith (Ginn & Co, 1923). MR 1520803 article

[289]L. E. Dickson: “Algèbres nouvelles de division” [New division algebras], C. R. Acad. Sci. Paris 181 (1925), pp. 836–838. See also Trans. Am. Math. Soc. 28:2 (1926) and Bull. Am. Math. Soc. 34:5 (1928). JFM 51.0121.01 article

[290]L. E. Dickson: “Book review: Lehrbuch der Algebra, verfasst mit Benutzung von Heinrich Webers gleichnamigem Buche,” Bull. Am. Math. Soc. 31 : 7 (1925), pp. 372–373. Book by R. Fricke (Veiweg, 1924). MR 1561066 article

Robert Fricke	Related
Heinrich Webers	Related

[291]L. E. Dickson: “Resolvent sextics of quintic equations,” Bull. Am. Math. Soc. 31 : 9–10 (1925), pp. 515–523. MR 1561108 JFM 51.0093.04 article

[292]L. E. Dickson: “Quadratic forms which represent all integers,” Proceedings of the National Academy of Sciences of the United States of America 12 (December 1926), pp. 756–757. JFM 52.0145.03 article

[293]L. E. Dickson: “All integral solutions of \( ax^2+bxy+cy^2=w_1 w_2\cdots w_n \),” Bull. Am. Math. Soc. 32 : 6 (1926), pp. 644–648. An abstract was published in Bull. Am. Math. Soc. 32:6 (1926). MR 1561282 JFM 52.0147.04 article

Leonard Eugene Dickson

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