Celebratio Mathematica — Dickson

Leonard Eugene Dickson 1874–1954

by A. A. Albert

Leonard Eugene Dickson was born in Independence, Iowa, on January 22, 1874. He was a brilliant undergraduate at the University of Texas, receiving his B.S. degree as valedictorian of his class in 1893. He was a chemist with the Texas Biological Survey from 1892 to 1893. He served as a teaching fellow at the University of Texas, receiving the M.A. degree in 1894. He held a fellowship at the University of Chicago from 1894 to 1896, and was awarded its first Ph.D. in mathematics in 1896. He spent the year 1896–1897 in Leipzig and Paris, was instructor in mathematics at the University of California 1897–1899, associate professor at Texas 1899–1900, assistant professor at Chicago 1900–1907, associate professor 1907–1910, and professor in 1910. He was appointed to the Eliakim Hastings Moore Distinguished Professorship in 1928, and became professor emeritus in 1939. He served as visiting professor at the University of California in 1914, 1918, and 1922.

Professor Dickson was awarded the $1,000 A.A.A.S. Prize in 1924 for his work on the arithmetics of algebras. He was awarded the Cole Prize of the American Mathematical Society in 1928 for his book Albegren und ihre Zahlentheorie [42]. He served as editor of the Monthly 1902–1908, and the Transactions from 1911 to 1916, and he was president of the American Mathematical Society from 1916–1918. He was elected to membership in the National Academy of Sciences in 1913, and was a member of the American Philosophical Society, the American Academy of Arts and Sciences, and the London Mathematical Society. He was also a foreign member of the Academy of the Institute of France, and an honorary member of the Czechoslovakian Union of Mathematics and Physics. He was awarded the honorary Sc.D. degree by Harvard in 1936 and Princeton in 1941.

Professor Dickson died in Texas on January 17, 1954.

Dickson was one of our most prolific mathematicians. His bibliography (prepared by Mr. Richard Block, a student at the University of Chicago) contains 285 titles. Of these, eighteen are books, one a joint book with Miller and Blichfeldt [33]. One of the books is his major three-volume History of the theory of numbers [35], [36], [39], which would be a life’s work by itself for a more ordinary man.

Dickson was an inspiring teacher. He supervised the doctorate dissertations of at least fifty-five Chicago Ph.D.s. He helped his students to get started in research after the Ph.D. and his books had a world-wide influence in stimulating research.

We now pass on to a brief discussion of Dickson’s research.

1. Linear groups

Dickson’s first major research effort was a study of finite linear groups. All but seven of his first forty-three papers were on that subject, and this portion of his work led to his famous first book [3]. The linear groups which had been investigated by Galois, Jordan, and Serret were all groups over the fields of $ p $ elements. Dickson generalized their results to linear groups over an arbitrary finite field. He obtained many new systems of simple groups, and he closed his book with a still valuable summary of the known systems of simple groups.

Dickson’s work on linear groups continued until 1908, and he wrote about forty-four additional papers on the subject. In these later papers he studied the isomorphism of certain simple groups and questions about the existence of certain types of subgroups. He also derived a number of theorems on infinite linear groups.

2. Finite fields and Chevalley’s theorem

In [3]. Dickson gave the first extensive exposition of the theory of finite fields. He applied his deep knowledge of that subject not only to linear groups but to other problems which we shall discuss later. He studied irreducibility questions over a finite field in [11], the Galois theory in [10], and forms whose values are squares in [19]. His knowledge of the role of the non-null form was shown in [20]. In [16] Dickson made the following statement: “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.” This result was first proved by C. Chevalley in his paper Demonstration d’une hypothese de M. Artin, Hamb. Abh, vol. 11 (1935), pp. 73–75. At least the conjecture should have been attributed to Dickson, who actually proved the theorem for $ m = 2, 3 $.

3. Invariants

Several of Dickson’s early papers were concerned with the problems of the algebraic geometry of his time. For example, see [1], [5], [4]. This work led naturally to his study of algebraic invariants, and his interest in finite fields to modular invariants. He wrote a basic paper on the latter subject in [18], and many other papers on the subject. In these papers he devoted a great deal of space to the details of a number of special cases. His book [24], on the classical theory of algebraic invariants, was published in 1914, the year after the appearance of his colloquium lectures. His amazing productivity is attested to by the fact that he also published his book [25] on linear algebras in 1914.

4. Algebras

Dickson played a major role in research on linear algebras. He began with a study of finite division algebras in [6], [7], [9] and [8]. In these papers he determined all three- and four-dimensional (nonassociative) division algebras over a field of characteristic not two, a set of algebras of dimension six, and a method for constructing algebras of dimension $ mk $ with a subfield of the dimension $ m $. In [15] he related the theory of ternary cubic forms to the theory of three-dimensional division algebras. His last paper on non-associative algebras [47] appeared in 1937 and contained basic results on algebras of degree two.

Reference has already been made to Dickson’s first book on linear algebras. In that text he gave a proof of his result that a real Cayley division algebra is actually a division algebra. He presented the Cartan theory of linear associative algebras rather than the Wedderburn theory, but stated the results of the latter theory in his closing chapter without proofs. The present value of this book is enhanced by numerous bibliographical references.

Dickson defined cyclic algebras in a Bulletin abstract of vol. 12 (1905–1906). His paper [21] on the subject did not appear until 1912, where he presented a study of algebras of dimension 16.

Dickson’s work on the arithmetics of algebras first appeared in [37]. His major work on the subject of arithmetics was presented in [38] where he also gave an exposition of the Wedderburn theory. See also [44] and [45].

The text [42] is a German version of [38]. However, the new version also contains the results on crossed product algebras which had been published in [46], and contains many other items of importance.

5. Theory of numbers

Dickson always said that mathematics is the queen of the sciences, and that the theory of numbers is the queen of mathematics. He also stated that he had always wished to work in the theory of numbers and that he wrote his monumental three-volume History of the theory of numbers [35], [36], [39] so that he could know all of the work which had been done in the subject. His first paper [2] contained a generalization of the elementary Fermat theorem which arose in connection with finite-field theory. He was interested in the existence of perfect numbers, and wrote [23] and [22] on the related topic of abundant numbers. His interest in Fermat’s last theorem appears in [34], [12], [14], [13] and [17]. During 1926–1930 he spent most of his energy on research in the arithmetic theory of quadratic forms, in particular on universal forms.

Dickson’s interest in additive number theory began in 1927 with [43]. He wrote many papers on the subject during the remainder of his life. The analytic results of Vinogradov gave Dickson the hope of proving the so-called ideal Waring theorem. This he did in a long series of papers. His final result is an almost complete verification of the conjecture made by J. A. Euler in 1772. That conjecture stated that every positive integer is a sum of $ J $ $ n $-th powers, where we write $ 3^n = 2^n q + r $, $ \,J = 2^n > r > 0 $, and $ J = 2^n + q - 2 $. Dickson showed that if $ n > 6 $ this value is correct unless $ q+ r+3 > 2^n $. It is still not known whether or not this last inequality is possible, but if it does occur the number $ g(n) $ of such $ n $-th powers required to represent all integers is $ J + f $ or $ J+f-1 $, according as $ fq + f+ q = 2^n $ or $ fq + f+q > 2^n $, where $ f $ is the greatest integer in $ (4/3)^n $ .

6. Miscellaneous

We close by mentioning Dickson’s interest in the theory of matrices which is best illustrated by his text, Modern algebraic theories [41]. His geometric work in [26], [30], [29], [32], [27], [31] and [28] must also be mentioned, as well as his interesting monograph [40] on differential equations from the Lie-group standpoint.

Works

[1]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

[2]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.

[3]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

[4]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

[5]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.

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

[7]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

[8]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.

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

[10]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

[11]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

[12]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

[13]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

[14]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

[15]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.

[16]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.

[17]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

[18]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.

[19]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

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

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

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

[23]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

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

[25]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

[26]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

[27]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.

[28]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.

[29]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.

[30]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.

[31]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

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

[33]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

[34]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

[35]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

[36]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

[37]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.

[38]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

[39]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

[40]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.

[41]L. E. Dickson: Modern algebraic theories. B. H. Sanborn & Co. (Chicago), 1926. Republished as Algebraic theories (1959). JFM 52.0094.01 book

[42]L. E. Dickson: Algebren und ihre Zahlentheorie [Algebras and their arithmetics]. Veröffentlichungen der Schweizerischen Mathematischen Gesellschaft 4. Orell Füssli (Zürich), 1927. Translation of completely revised and extended manuscript, with contribution on ideal theory from Andreas Speiser. German translation of Algebras and their arithmetics (1923). JFM 53.0112.01 book

[43]L. E. Dickson: “Extensions of Waring’s theorem on nine cubes,” Amer. Math. Mon. 34 : 4 (April 1927), pp. 177–183. MR 1521139 JFM 53.0134.02 article

[44]L. E. Dickson: “Outline of the theory to date of the arithmetics of algebras,” pp. 95–102 in Proceedings of the International Mathematical Congress, 1924 (Toronto, 11–16 August, 1924), vol. 1. Edited by J. C. Fields. University of Toronto Press, 1928. JFM 54.0160.03 incollection

[45]L. E. Dickson: “Further development of the theory of arithmetics of algebras,” pp. 173–184 in Proceedings of the International Mathematical Congress, 1924 (Toronto, 11–16 August, 1924), vol. 1. Edited by J. C. Fields. University of Toronto Press, 1928. JFM 54.0161.01 incollection

[46]L. E. Dickson: “New division algebras,” Bull. Am. Math. Soc. 34 : 5 (1928), pp. 555–560. See also Trans. Am. Math. Soc. 28:2 (1926) and C. R. Acad. Sci. Paris 181 (1925). MR 1561617 JFM 54.0161.03 article

No technical acquaintance with linear algebras is presupposed in this note. We consider only linear algebras for which multiplication is associative. As with quaternions, an algebra $ A $ is called a division algebra if every element $ \neq 0 $ of $ A $ has an inverse in $ A $. A division algebra $ A $ over a field $ F $ is called normal if the numbers of $ F $ are the only elements of $ A $ which are commutative with every element of $ A $.

In a paper recently offered to the Transactions of this Society, A. A. Albert determined all normal division algebras of order $ {}16 $ and found a new type. The object of this note is to derive from mild assumptions the corresponding type of normal division algebras $ A $ of order $ {}4p^2 $, where $ p $ is a prime.

[47]L. E. Dickson: “Linear algebras with associativity not assumed,” Duke Math. J. 1 : 2 (1935), pp. 113–125. MR 1545870 JFM 61.0125.01 Zbl 0012.14801 article

The complete struture of linear associative algebras was known to depend upon the division algebras. When the reference field $ F $ is an algebraic field, H. Hasse has recently proved that every normal division algebra is cyclic. This perfection of the theory of associative algebras justifies attention to non-associative algebras.

Known examples of non-associative division algebras are Cayley’s algebra of order $ {}8 $, and the writer’s [1914, p. 14, p. 69 (p. 17 for the characteristic equations); 1907–1908, p. 169; 1906a; 1906b] commutative algebras of orders $ {}3 $ and $ {}2n $ (§15). Many new division algebras of order $ {}4 $ are given here by Theorems 2 and 3.

In §§7–11 we determine all types of algebras of order $ {}3 $ having a principal unit (or modulus) denoted by $ {}1 $. Except for special values of the parameters, these algebras are simple. It is known that every associative simple algebra of order $ {}3 $ is a division algebra.