Celebratio Mathematica — Albert

[1] A. A. Albert: A determination of all associative algebras in two, three and four units over a nonmodular field \( \frak{F} \). M.S. Thesis, University of Chicago, 1927. phdthesis

[2]A. A. Albert: Algebras and their radicals and division algebras. Ph.D. thesis, University of Chicago, 1928. Advised by L. E. Dickson. MR 2611246 phdthesis

[3]A. A. Albert: “Normal division algebras satisfying mild assumptions,” Proc. Nat. Acad. Sci. U.S.A. 14 : 12 (December 1928), pp. 904–906. JFM 54.0161.04 article

[4]A. A. Albert: “The group of the rank equation of any normal division algebra,” Proc. Natl. Acad. Sci. USA 14 : 12 (December 1928), pp. 906–907. JFM 54.0161.05 article

[5]A. A. Albert: “A determination of all normal division algebras in sixteen units,” Bull. Am. Math. Soc. 34 : 4 (1928), pp. 410–411. Abstract only; paper in Trans. Am. Math. Soc. 31:2 (1929). JFM 54.0166.16 article

[6]A. A. Albert: “On the structure of normal division algebras,” Ann. Math. (2) 30 : 1–4 (1928–1929), pp. 322–338. An abstract was published in Bull. Am. Math. Soc. 35:2 (1929). MR 1502886 JFM 55.0682.01 article

The chief outstanding problem in the theory of linear associative algebras is the determination of all normal division algebras. In the present paper we shall add to the present meager knowledge of the structure of such algebras by proving certain basic theorems on the structure of algebras satisfying mild postulates. We shall also, in the exposition of this theory, prove certain theorems tactily assumed by writers in the theory of division algebras, and define new terms to be used here and in subsequent papers.

We shall consider normal linear associative division algebras over a non-modular field \( F \) and shall speak of them, simply, as normal division algebras. The general non-modular reference field \( F \) shall be understood unless the contrary is explicitly stated.

[7]A. A. Albert: “Normal division algebras in \( 4p^2 \) units, \( p \) an odd prime,” Ann. Math. (2) 30 : 1–4 (1928–1929), pp. 583–590. MR 1502907 JFM 55.0683.01 article

We shall consider normal associative division algebras in \( 4p^2 \) units, over any non-modular field \( F \), satisfying special assumptions. We shall utilize the definitions and theorems of a previous paper [1929], referring to the theorems by numbers 1 to 29 and shall number the theorems of this paper beginning with 30.

In the paper above referred to we have defined “known algebras” and “the type of an irreducible equation, of an element of a normal division algebra, and of a normal division algebra”.

L. E. Dickson [1926] has found the structure of all normal division algebras in \( n^2 \) units over \( F \), of type \( R_n \), that is, normal division algebras containing an element \( x \) whose minimum equation is of degree \( n \) and has the property that all of its scalar roots are rational functions, with coefficients in \( F \), of one of them. He has also determined and constructed the algebras when the minimum equation of this \( x \) has a solvable group. These algebras are easily seen to be known algebras under our definition and the resulting criterion shall be utilized here.

[8]A. A. Albert: “The structure of any algebra which is a direct product of rational generalized quaternion division algebras,” Ann. Math. (2) 30 : 1–4 (1928–1929), pp. 621–625. MR 1502911 JFM 55.0683.02 article

[9]A. A. Albert: “On the rank equation of any normal division algebra,” Bull. Am. Math. Soc. 35 : 3 (1929), pp. 335–338. MR 1561740 JFM 55.0090.03 article

The different types of normal division algebras which have been discovered up to the present depend upon equations with different groups. It has been thought that, as the rank equation of an algebra is invariant under a change of basal units, the groups of the rank equations of these various types of algebras might serve to show their non-equivalence. This notion is shown to be false here, as the group of the rank equation of any normal division algebra is the symmetric group. In proving this theorem a new theorem in the Hubert theory of an irreducible polynomial whose coefficients are rational functions, with coefficients in any infinite field \( K \), of several parameters is developed.

[10]A. A. Albert: “A determination of all normal division algebras in sixteen units,” Trans. Am. Math. Soc. 31 : 2 (1929), pp. 253–260. An abstract was published in Bull. Am. Math. Soc. 34:4 (1928). MR 1501481 JFM 55.0090.04 article

The chief outstanding problem in the theory of linear associative algebras over an infinite field \( F \) is the determination of all division algebras. This problem is equivalent to that of the determination of all normal division algebras, or algebras \( A \) in which the only elements of \( A \) commutative with every element of \( A \) are the quantities of its reference field \( F \).

The order of a normal division algebra is the square of an integer. All normal division algebras in \( 1^2 \), \( 2^2 \) [Dickson 1927, p. 46], and \( 3^2 \) [Wedderburn 1921, p. 132] units have been determined. In this paper all normal division algebras in \( 4^2 \) units, the next case, are determined and shown to be the algebras of Cecioni [1923].

[11]A. A. Albert: “The rank function of any simple algebra,” Proc. Natl. Acad. Sci. U.S.A. 15 : 4 (April 1929), pp. 372–376. An abstract was published in Bull. Am. Math. Soc. 35:4 (1929). JFM 55.0683.03 article

[12]A. A. Albert: “On the structure of normal division algebras,” Bull. Am. Math. Soc. 35 : 2 (1929), pp. 182–183. Abstract only; paper in Ann. Math. 30:1–4 (1928–1929). JFM 55.0092.03 article

[13]A. A. Albert: “The rank function of any simple algebra,” Bull. Am. Math. Soc. 35 : 4 (1929), pp. 441. Abstract only; paper in Proc. Natl. Acad. Sci. U.S.A. 15:4 (1929). JFM 55.0092.05 article

[14]A. A. Albert: “A necessary and sufficient condition for the non-equivalence of any two rational generalized quaternion division algebras,” Bull. Am. Math. Soc. 36 : 8 (1930), pp. 535–540. An abstract was published in Bull. Am. Math. Soc. 36:3 (1930). MR 1561990 JFM 56.0145.03 article

[15]A. A. Albert: “A construction of all non-commutative rational division algebras of order eight,” Bull. Am. Math. Soc. 36 (1930), pp. 214. Article only; paper in Ann. Math. 31:4 (1930). JFM 56.0152.24 article

By rational algebras is meant algebras over the field \( R \) of all rational numbers. All rational generalized quaternion division algebras and all rational cyclic division algebras of order sixteen have been constructed [Albert 1930]. Also a type of cyclic division algebra of order nine has been given [Dickson 1923, pp. 68–71]. In the present paper the problem of the construction of all non-commutative rational division algebras of order eight is considered and solved, necessary and sufficient conditions being given that the algebras constructed be division algebras. These conditions are explicit diophantine conditions on the constants of the general algebra in one of two canonical forms.

[16]A. A. Albert: “New results in the theory of normal division algebras,” Trans. Am. Math. Soc. 32 : 2 (1930), pp. 171–195. An abstract was published in Bull. Am. Math. Soc. 36:3 (1930). MR 1501532 JFM 56.0146.01 article

The author shows that in every normal division algebra \( A \) in sixteen units over \( R \), the field of all rational numbers, there are eight integer parameters \( \rho, \sigma, \gamma_1,\dots,\gamma_6 \), where neither \( \rho \), \( \sigma \), nor \( \rho\sigma \) is a rational square. Algebra \( A \) is associative if and only if \[ (\gamma_5^2 - \gamma_6^2\sigma\rho) = (\gamma_1^2-\gamma_2^2\rho)(\gamma_3^2 - \gamma_4^2\sigma) \] and is a division algebra if and only if the ternary quadratic form \[ \lambda_1^2 - \sigma\lambda_2^2 - (\gamma_1^2 - \gamma_2^2\rho)\lambda_3^2 ,\] in the integer variables \( \lambda_1 \), \( \lambda_2 \), \( \lambda_3 \) is not a null form. As all ternary quadratic null forms are known, this amounts to a construction of all normal division algebras of order sixteen over \( R \) in terms of the solutions of a single diophantine equation which satisfy a single quadratic residue condition. Necessary and sufficient algebraic conditions that \( A \) be a cyclic algebra are given, and it is shown that all of the algebras constructed by Cecioni are cyclic. An algebra \( A \) is shown to be non-cyclic if and only if two quartic forms with coefficients polynomials in \( \rho \), \( \sigma \), \( \gamma_1,\dots,\gamma_6 \) are non-null forms.

[17]A. A. Albert: “A determination of the integers of all cubic fields,” Ann. Math. (2) 31 : 4 (October 1930), pp. 550–566. An abstract was published in Bull. Am. Math. Soc. 36:3 (1930). MR 1502961 JFM 56.0870.01 article

[18]A. A. Albert: “A note on an important theorem on normal division algebras,” Bull. Am. Math. Soc. 36 : 10 (1930), pp. 649–650. MR 1562029 JFM 56.0145.04 article

[19]A. A. Albert: “Determination of all normal division algebras in thirty-six units of type \( R_2 \),” Am. J. Math. 52 : 2 (1930), pp. 283–292. MR 1507918 JFM 56.0145.06 article

An element \( x \) of a normal division algebra \( A \) in \( n^2 \) units over a non-modular field \( F \) is said to have grade \( r \) if its minimum equation has degree \( r \). If there exist \( k \) distinct elements \( x \), \( \theta_1(x),\dots,\theta_{k-1}(x) \) which are polynomials in \( x \) and satisfy its minimum equation then \( x \) is said to have type \( R_k \). Algebra \( A \) in \( n^2 \) units has type \( R_k \) if it contains an element of grade \( n \) and type \( R_k \).

In all of the papers [Wedderburn 1921; Albert 1929a] on the determination of normal division algebras the algebras were first shown to be of type \( R_2 \) and then proved to be known algebras. For algebras \( A \) in \( n^2 \) units, \( n \) a prime, the assumption that \( A \) has type \( R_2 \) implies, by an almost trivial argument, that \( A \) has type \( R_n \) and is a cyclic algebra. But when \( n \) is composite the problem of showing \( A \) to be known is difficult even under the assumption that \( A \) has type \( R_2 \). We consider here the case \( n=6 \) and show that all normal division algebras in thirty-six units of type \( R_2 \) are of type \( R_3 \) and are known in the broad sense recently defined [Albert 1929b]. A necessary and sufficient condition that the known algebras be of type \( R_6 \) and hence of the kind constructed by L. E. Dickson is also given.

[20]A. A. Albert: “The non-existence of pure Riemann matrices with normal multiplication algebras of order sixteen,” Ann. Math. (2) 31 : 3 (1930), pp. 375–380. An abstract was published in Bull. Am. Math. Soc. 36:3 (1930). MR 1502951 JFM 56.0869.02 article

The most interesting question in the theory of Riemann matrices is that of the existence of pure Riemann matrices with non-commutative multiplication algebras. In a recent paper the author found necessary and sufficient conditions on a normal division algebra that it be a multiplication algebra of a pure Riemann matrix and gave examples of matrices whose algebras are generalized quaternion algebras and of matrices whose algebras are generalized quaternion algebras over a quadratic field. In the present paper it is shown that there exist no pure Riemann matrices whose multiplication algebras are normal division algebras over the field of all rational numbers.

[21]A. A. Albert: “The integers of normal quartic fields,” Ann. Math. (2) 31 : 3 (July 1930), pp. 381–418. MR 1502952 JFM 56.0870.02 article

The integral bases of all relative quadratic fields \( K(\sqrt{\mu}) \) where \( \mu \) is in \( K = R(\sqrt{m}) \) have been found [Sommer 1907] by the theory of ideals. But these bases are given in terms of factorization of two and \( \mu \) into prime ideals, the solution of a quadratic congruence modulo ideals and further complicated processes. The formulae are inexplicit and are almost useless for application to special cases, a situation not at all like that of the case of quadratic fields.

We consider here all normal quartic fields \( R(i) \) where \( R \) is the field of all rational numbers and \( i \) is a root of a quartic equation with rational coefficients and either the cyclic group or the “Vierergruppe” with respect to \( R \). All such fields are relative quadratic fields. But here, utilizing the known canonical forms of such quartics and strictly elementary methods, we obtain explicit formulae for the bases of the integers of any normal quartic field in terms of the coefficients of the corresponding quartic equation.

[22]A. A. Albert: “On the structure of pure Riemann matrices with non-commutative multiplication algebras,” Proc. Natl. Acad. Sci. U.S.A. 16 : 4 (April 1930), pp. 308–312. JFM 56.0329.08 article

[23]A. A. Albert: “New results in the theory of normal division algebras,” Bull. Am. Math. Soc. 36 (1930), pp. 198–199. Abstract only; paper in Trans. Am. Math. Soc. 32:2 (1930). JFM 56.0152.19 article

[24]A. A. Albert: “On direct products, cyclic division algebras, and pure Riemann matrices,” Proc. Natl. Acad. Sci. U.S.A. 16 : 4 (April 1930), pp. 313–315. See also Trans. Am. Math. Soc. 33: 1 (1931). JFM 56.0869.01 article

[25]A. A. Albert: “On the Wedderburn norm condition for cyclic algebras,” Bull. Am. Math. Soc. 36 : 11 (1930), pp. 804. Abstract only; paper in Bull. Am. Math. Soc. 37:4 (1931). JFM 56.0153.01 article

[26]A. A. Albert: “A necessary and sufficient condition for the non-equivalence of any two rational generalized quaternion division algebras,” Bull. Am. Math. Soc. 36 : 3 (1930), pp. 213. Abstract only; paper in Bull. Am. Math. Soc. 36:8 (1930). JFM 56.0152.23 article

[27]A. A. Albert: “A determination of the integers of all cubic fields,” Bull. Am. Math. Soc. 36 : 3 (1930), pp. 199. Abstract only; paper in Ann. Math. 31:4 (1930). JFM 56.0152.22 article

[28]A. A. Albert: “The non-existence of pure Riemann matrices with normal multiplication algebras of order sixteen,” Bull. Am. Math. Soc. 36 : 3 (1930), pp. 199. Abstract only; paper in Ann. Math. 31:3 (1930). JFM 56.0152.20 article

[29]A. A. Albert: “Division algebras over an algebraic field,” Bull. Am. Math. Soc. 37 : 10 (1931), pp. 777–784. An abstract was published in Bull. Am. Math. Soc. 37:7 (1931). MR 1562255 JFM 57.0161.01 Zbl 0003.05303 article

H. Hasse has given a rigorous treatment of the theory of quadratic null forms over the field \( R \) of all rational numbers. He used the theory of \( p \)-adic numbers and readily extended his methods to obtain complete results on quadratic forms over any algebraic number field [1923] and hence, by a simple isomorphism, over any field \( R(\theta) \), where \( \theta \) is any quantity satisfying an equation with coefficients in \( R \) and irreducible in \( R \). Hasse has also used the fundamental principle of his quadratic form theory to prove several important theorems on cyclic (Dickson) algebras [1932].

I have recently obtained theorems on rational division algebras by the use of A. Meyer’s theorem that every indefinite quadratic form, with rational coefficients, in five or more variables is a null form. But now Hasse’s theorems make the extension to algebras over algebraic fields \( R(\theta) \) almost immediate. In particular it is shown here that the direct product of any two generalized quaternion algebras over \( R(\theta) \) is never a division algebra, and that a sufficient condition that a normal division algebra of order sixteen over \( R(\theta) \) be a cyclic algebra is that it contain a quantity \( x \) not in \( R(\theta) \) such that \( x^2 = \Delta_1^2 + \Delta_2^2 \) with \( \Delta_1 \) and \( \Delta_2 \) in \( R(\theta) \).

Hasse’s new theorems on cyclic algebras are also used here to obtain an alternative proof of the above theorem on generalized quaternion algebras. In fact I show here that a necessary and sufficient condition that a direct product of any two normal division algebras over a field \( R(\theta) \) be a division algebra is that their orders be relatively prime.

[30]A. A. Albert: “Errata: ‘On direct products, cyclic division algebras, and pure Riemann matrices’,” Trans. Am. Math. Soc. 33 : 4 (October 1931), pp. 999. Errata for paper published in Trans. Am. Math. Soc. 33:1 (1931). MR 1500511 article

[31]A. A. Albert: “The structure of matrices with any normal division algebra of multiplications,” Ann. Math. (2) 32 : 1 (January 1931), pp. 131–148. MR 1502987 JFM 57.0162.01 Zbl 0001.26603 article

The outstanding problem in the theory of Riemann matrices is the determination of all pure Riemann matrices. The chief sub-problem has been that of finding the structure of a pure Rieman matrix with a given multiplication algebra. This problem was solved for the case of fields by S. Lefschetz, was reduced essentially to the case of nornal division algebras by the author, and was solved for the case of “known” normal division algebras by the author [1930].

In the present paper the author defines algebras of multiplications of matrices not necessarily Riemann matrices and finds necessary and sufficient conditions that a matrix have a given algebra of multiplications. By adding the conditions that a given matrix be a Riemann matrix having no multiplication not in the given algebra, the author completely determines the structure of all pure Riemann matrices with a given multiplication algebra.

[32]A. A. Albert: “A note on cyclic algebras of order sixteen,” Bull. Am. Math. Soc. 37 : 10 (1931), pp. 727–730. An abstract was published in Bull. Am. Math. Soc. 37:7 (1931). MR 1562244 JFM 57.0160.03 Zbl 0003.05302 article

In a recent paper [1931] I considered cyclic (Dickson) algebras of order sixteen generated by a cyclic quartic field \( Z \) and a quantity \( \gamma \) in the reference field \( F \). It was proved there that if the algebra \( A \) were a division algebra and if \( \gamma^2 \) were the norm of a quantity of \( Z \), so that the Wedderburn norm condition would not be satisfied, then \( A \) would be the direct product of two generalized quaternion algebras. It was not proved, however, that such division algebras existed.

R. Brauer has recently [1930] proved that there exist normal division algebras which are direct products of two generalized quaternion algebras. In the present note I give an example of a cyclic algebra over the Brauer reference field for which the norm condition is not satisfied, therefore completing the theory of the previous paper.

[33]A. A. Albert: “On direct products, cyclic division algebras, and pure Riemann matrices,” Trans. Am. Math. Soc. 33 : 1 (1931), pp. 219–234. Errata published in Trans. Am. Math. Soc. 33:4 (1931). See also Proc. Natl. Acad. Sci. U.S.A. 16:4 (1930). MR 1501586 JFM 57.0158.03 Zbl 0001.11602 article

The present paper is the result of a consideration of several related topics in the theory of linear associative algebras and the application of the results obtained to the theory of Riemann matrices. We first consider a linear algebra problem of great importance in its application to Riemann matrix theory, the question as to when a normal division algebra of order \( n^2 \) over \( F \) is representable by an algebra of \( m \)-rowed square matrices with elements in \( F \). It is shown that this is possible if and only if \( n^2 \) divides \( m \) and this is applied to prove that.

The multiplication index \( h \) of a pure Riemann matrix of genus \( p \) is a divisor of \( 2p \).

The algebras called cyclic (Dickson) algebras are the simplest normal division algebras structurally. J. H. M. Wedderburn has given sufficient conditions that constructed cyclic algebras be division algebras but it seems to have been overlooked that these conditions have not been shown to be necessary. In the fundamental problem of the construction of all such division algebras necessary and sufficient conditions are of course needed. The question of the necessity of the Wedderburn conditions is considered here and the results applied to obtain what seems a remarkable restriction on the types of algebras which may be the multiplication algebras of pure Riemann matrices. Also certain general theorems on direct products of normal simple algebras are proved and the conclusions used to reduce the problem of the construction of cyclic division algebras to the case where the order of the algebra is a power of a single prime.

[34]A. A. Albert: “On normal division algebras of type \( R \) in thirty-six units,” Trans. Am. Math. Soc. 33 : 1 (January 1931), pp. 235–243. MR 1501587 JFM 57.0160.01 Zbl 0001.11701 article

A normal division algebra in \( n^2 \) units over a modular field \( F \) is of type \( R \) if it contains a quantity \( i \) whose minimum equation with respect to \( F \), \( \phi(\omega) = 0 \), has degree \( n \) and \( n \) distinct roots which are polynomials in \( i \) with coefficients in \( F \). Algebras of type \( R \) occupy a central position in the theory of division algebras as they are the only normal division algebras whose structure is known, and all division algebras of order less than twenty-five are expressible as algebras of type \( R \).

The normal division algebras \( D \) whose structure is the simplest are those for the case where \( \phi(\omega)=0 \) has the cyclic group with respect to \( F \). When \( n \) is six and \( \phi(\omega)=0 \) is cyclic, \( D \) is expressible as the direct product of a generalized quaternion division algebra and a cyclic division algebra of order nine, while conversely every such direct product is a cyclic division algebra of order thirty-six. The group of \( \phi(\omega) = 0 \) is evidently regular and hence the only other type of equation to be considered for algebras of order thirty-six and type \( R \) is one which as the single non-cyclic, non-abelian regular group on six letters, a case giving a very complicated algebra.

It has never been demonstrated that there exist normal division algebras which are not cyclic algebras. The author showed, in a recent paper [1930], that the algebras which had been constructed by F. Cecioni [1923] and which were based on a non-cyclic quartic were cyclic algebras. We show here that all normal division algebras of type \( R \) in thirty-six units are cyclic algebras.

[35]A. A. Albert: “On the Wedderburn norm condition for cyclic algebras,” Bull. Am. Math. Soc. 37 : 4 (1931), pp. 301–312. An abstract was published in Bull. Am. Math. Soc. 36:11 (1930). MR 1562142 JFM 57.0160.02 Zbl 0001.26701 article

In the present paper cyclic algebras of order sixteen with the corresponding cyclic quartic in its canonical form [Garver 1928] \[ \phi(\omega) \equiv \omega^4 + 2\nu(1+\Delta^2)\omega^2 + \nu^2\Delta^2(1+\Delta^2) = 0 \] such that \( \nu \) and \( \Delta \) are in \( F \), and \( \tau = 1+\Delta^2 \) is not the square of any quantity of \( F \), are considered. The norm \( N(a) \) of a polynomial in \( i \) is a rather complicated quartic form in four variables, yet we can secure the result that \( \gamma^2 = N(a) \) if and only if \( \gamma = \alpha^2 - \beta^2\tau \) for \( \alpha \) and \( \beta \) in \( F \), a curious property of cyclic quartic fields. When the above equation is satisfied the algebra \( A \) is expressible as a direct product of two generalized quaternion algebras. Necessary and sufficient conditions are secured that our algebras \( A \) of order sixteen be division algebras, and it is shown that for the particularly interesting case where \( F \) is the field of all rational numbers the Wedderburn condition is necessary as well as sufficient.

[36]A. A. Albert: “On direct products,” Trans. Am. Math. Soc. 33 : 3 (1931), pp. 690–711. An abstract was published in Bull. Am. Math. Soc. 37:5 (1931). MR 1501610 JFM 57.0159.01 Zbl 0002.24605 article

The principal contribution of this article is a set of theorems on the structure of the direct product of a normal division algebra \( A \) over \( F \) and an algebraic field \( F(\eta) \), with applications of the Galois theory of equations. The theorems are useful new tools for research on normal division algebras. In particular it is shown that it is possible to extend the reference field \( F \) of any normal division algebra \( A \) of order \( p^2 \) over \( F \), \( p \) a prime, such that \[ A^{\prime} = A\times F(\eta) \] is a cyclic normal divison algebra over \( F(\eta) \). A new proof is given of a little known theorem of R. Brauer [1929] which reduces the problem of determinining all normal division algebras of order \( n^2 \) over \( F \) to the case where \( n \) is a power of a prime.

[37]A. A. Albert: “Normal division algebras of order \( 2^{2^m} \),” Proc. Natl. Acad. Sci. USA 17 : 6 (June 1931), pp. 389–392. JFM 57.0161.02 Zbl 0002.11502 article

[38]A. A. Albert: “The structure of pure Riemann matrices with non-commutative multiplication algebras,” Rend. Circ. Mat. Palermo 55 : 1 (1931), pp. 57–115. JFM 57.0161.03 Zbl 0001.26602 article

The chief outstanding problem in the theory of Riemann matrices is the determination of the structure of all pure Riemann matrices \( \omega \) with a given non-commutative division algebra \( B \) as its multiplication algebra \( D \).

In the present paper we shall consider pure Riemann matrices over any sub-field of the field of all real numbers, an extension of ordinary theory which is given in terms of the field of all rational numbers. This extension is made necessary when we consider submatrices of ordinary pure Riemann matrices. We give a rigorous algebraic proof of Rosatti’s main results [1929] and obtain his theorems, not in terms of unknown sub-algebras of given algebras \( B \) but actually in terms of the central field of \( B \) and its rank function. The criteria we obtain are in terms of concrete properties of the algebras to be considered and are therefore immediately applicable to restrict the class of algebras which may possibly be multiplication algebras.

[39]A. A. Albert: “A note on cyclic algebras of order sixteen,” Bull. Am. Math. Soc. 37 : 7 (1931), pp. 527–528. Abstract only; paper in Bull. Am. Math. Soc. 37:10 (1931). JFM 57.0172.01 article

[40]A. A. Albert: “On the construction of cyclic algebras with a given exponent,” Bull. Am. Math. Soc. 37 : 7 (1931), pp. 528. Abstract only; paper in Am. J. Math. 54:1 (1932). JFM 57.0172.02 article

[41]A. A. Albert: “The integers represented by sets of positive ternary quadratic forms,” Bull. Am. Math. Soc. 37 : 3 (1931), pp. 169. Abstract only; paper in Am. J. Math. 55:1–4 (1933). JFM 57.0199.08 article

[42]A. A. Albert: “On direct products,” Bull. Am. Math. Soc. 37 : 5 (1931), pp. 355. Abstract only; paper in Trans. Am. Math. Soc. 33:3 (1931). JFM 57.0171.22 article

[43]A. A. Albert: “Division algebras over an algebraic field,” Bull. Am. Math. Soc. 37 : 7 (1931), pp. 528. Abstract only; paper in Bull. Am. Math. Soc. 37:10 (1931). JFM 57.0172.03 article

[44]A. A. Albert: “Algebras of degree \( 2^e \) and pure Riemann matrices,” Ann. Math. (2) 33 : 2 (April 1932), pp. 311–318. MR 1503054 JFM 58.0143.02 Zbl 0004.10001 article

The most recent discoveries in the theory of linear associative algebras have been due to the discussion of a new invariant number, the exponent of a normal division algebra. If \( A \) is any normal division algebra of degree \( n \) (order \( n^2 \)) over \( F \) then the exponent of \( A \) is the least integer \( \rho \) such that \( A^{\rho} \) is a total matric algebra. H. Hasse has shown that the exponent of any cyclic algebra over an algebraic field \( \Omega \) is equal to its degree. In the present paper I consider the corresponding question for any normal division algebra of degree a power of two over an algebraic field and prove that here again the exponent is equal to the degree. In particular I obtain the important theorem that the only self-reciprocal normal division algebras over an algebraic field are algebras of degree one or two.

The principal problem in the closely related theories of the complex multiplication of Riemann matrices, singular correspondences between algebraic curves, algebraic correspondences on an abelian variety, is that of a determination of the multiplication algebra \( D \) of any pure Riemann matrix. This algebra is a normal division algebra over an algebraic field. Either this field \( \Omega = R(s) \) where \( R \) is the field of all rational numbers and \( s \) is a quantity the roots \( \sigma_1,\dots,\sigma_t \) of whose minimum equation are all real, or \( \Omega = R(q,s) \) where \( s \) is as before and \( q^2 = \mu(s) \) such that \( \mu(\sigma_j) < 0 \). I have called a Riemann matrix \( \omega \) of the first or second kind according as \( \Omega = R(s) \) or \( \Omega = R(q,s) \). As an immediate corollary of my theorem of self-reciprocal algebras I show here that the multiplication algebra of any Riemann matrix of the first kind is either a field \( R(s) \) or a generalized quaternion algebra over \( R(s) \).

[45]A. A. Albert: “Normal division algebras of degree four over an algebraic field,” Trans. Am. Math. Soc. 34 : 2 (1932), pp. 363–372. MR 1501642 JFM 58.0144.01 Zbl 0004.10002 article

[46]A. A. Albert: “On normal simple algebras,” Trans. Am. Math. Soc. 34 : 3 (1932), pp. 620–625. An abstract was published in Bull. Am. Math. Soc. 38:3. MR 1501653 JFM 58.1026.04 Zbl 0005.05002 article

Recently published theorems on the direct product of a normal division algebra \( D \) of degree \( m \) (order \( m^2 \)) over \( F \) and an algebraic field \( Z \) of degree (order) \( r \) over \( F \) have proved to be very important tools for research on division algebras. Of particular value is the use of an integer \( s \) called the index reduction factor of \( D\times Z \). In the present paper new light is thrown on the properties of \( s \) by a study of an integer \( q = q(Z,D) \) called the quotient index of \( Z \) and \( D \). This \( q \) is the least integer such that the direct product of \( D \) and a total matric algebra of degree \( q \) contains a sub-field equivalent to \( Z \). It is proved that \( r=sq \). The results obtained are also applied to prove an important conjecture of L. E. Dickson made by him in 1926, the so-called norm condition that a certain type of algebra be a division algebra.

[47]A. A. Albert: “On the construction of cyclic algebras with a given exponent,” Am. J. Math. 54 : 1 (1932), pp. 1–13. An abstract was published in Bull. Am. Math. Soc. 37:7 (1931). MR 1506868 JFM 58.0141.05 Zbl 0003.24405 article

[48]A. A. Albert and H. Hasse: “A determination of all normal division algebras over an algebraic number field,” Trans. Am. Math. Soc. 34 : 3 (1932), pp. 722–726. An abstract was published in Bull. Am. Math. Soc. 38:3 (1932). MR 1501659 Zbl 0005.05003 article

[49]A. A. Albert: “A construction of non-cyclic normal division algebras,” Bull. Am. Math. Soc. 38 : 6 (1932), pp. 449–456. An abstract was published in Bull. Am. Math. Soc. 38:3 (1932). MR 1562415 JFM 58.0143.01 Zbl 0005.00603 article

We know now that every normal division algebra over an algebraic number field is a cyclic (Dickson) algebra. This result was proved by highly refined arithmetic means and the proof cannot be extended to obtain a like result for algebras over a general field. The very important question of whether or not any non-cyclic algebras exist has thus remained unanswered up to the present.

I shall give a construction of non-cyclic algebras of order sixteen over a function field in this paper. These algebras will be proved to be normal envision algebras; they furnish the first example in the literature of linear associative algebras of division algebras definitely known to be not of the Dickson type.

[50]A. A. Albert: “A note on normal division algebras of order sixteen,” Bull. Am. Math. Soc. 38 : 10 (1932), pp. 703–706. An abstract was published in Bull. Am. Math. Soc. 38:7 (1932). MR 1562492 JFM 58.0141.03 Zbl 0005.34202 article

[51]A. A. Albert: “A note on normal division algebras of order sixteen,” Bull. Am. Math. Soc. 38 : 7 (1932), pp. 496. Abstract only; paper in Bull. Am. Math. Soc. 38:10 (1932). JFM 58.0153.05 article

[52]A. A. Albert: “On normal simple algebras,” Bull. Am. Math. Soc. 38 : 3 (1932), pp. 179–180. Abstract only; paper in Trans. Am. Math. Soc. 34:3 (1932). JFM 58.0153.01 article

[53]A. A. Albert and H. Hasse: “A determination of all normal division algebras over an algebraic number field,” Bull. Am. Math. Soc. 38 : 3 (1932), pp. 179. Abstract only; paper in Trans. Am. Math. Soc. 34:3 (1932). JFM 58.0153.02 article

[54]A. A. Albert: “A construction of non-cyclic normal division algebras,” Bull. Am. Math. Soc. 38 : 3 (1932), pp. 180–181. Abstract only; paper in Bull. Am. Math. Soc. 38:6 (1932). JFM 58.0153.03 article

[55]A. A. Albert: “The integers represented by sets of ternary quadratic forms,” Am. J. Math. 55 : 1–4 (1933), pp. 274–292. An abstract was published in Bull. Am. Math. Soc. 37:3 (1931). MR 1506964 JFM 59.0181.03 Zbl 0006.29004 article

One of the most interesting topics in the theory of numbers is the study of the question of what integers are represented by positive ternary quadratic forms. Few general theorems are known in this subject. In fact, as L. E. Dickson has indicated, most of the forms are irregular [1926-27].

In the present paper a consideration is made of a different type of problem yet one that throws a good deal of light on the above topic. The problem of determining all the integers represented by the set \( \Sigma(d) \) of all positive ternary quadratic forms of the same determinant \( d \) is studied here and a complete solution is obtained. Moreover the results have the following remarkably simple form.

[56]A. A. Albert: “Normal division algebras over algebraic number fields not of finite degree,” Bull. Am. Math. Soc. 39 : 10 (1933), pp. 746–749. An abstract was published in Bull. Am. Math. Soc. 39:9 (1933). MR 1562722 JFM 59.0159.02 Zbl 0008.00104 article

If \( R \) is the field of all rational numbers and if \( \xi_1,\dots,\xi_n \) are ordinary algebraic numbers, then the field \[ \Omega = R(\xi_1,\dots,\xi_n) \] of all rational functions with rational coefficients of \( \xi_1,\dots,\xi_n \) is an algebraic number field of finite degree (the maximum number of linearly independent quantities of \( \Omega \)) over \( R \). It has recently been proved [Albert and Hasse 1932] that every normal simple algebra over such a field \( \Omega \) is cyclic. In particular it has been shown that every normal division algebra of order \( n^2 \) (degree \( n \)) over \( \Omega \) is cyclic and has exponent \( n \).

In the present note I shall give an extension of the above results to normal division algebras over any algebraic number field \( \Lambda \). I shall prove that all normal division algebras over \( \Lambda \) are cyclic and with degree equal to exponent but shall give a trivial example showing that the theorem corresponding to the above on normal simple algebras is false. The problem of the equivalence of normal division algebras over \( \Lambda \) will also be discussed.

[57]A. A. Albert: “Cyclic fields of degree eight,” Trans. Am. Math. Soc. 35 : 4 (1933), pp. 949–964. An abstract was published in Bull. Am. Math. Soc. 39:1 (1933). MR 1501727 JFM 59.0158.03 Zbl 0008.00201 article

Let \( F \) be any non-modular field, \( C \) be an algebraic extension of degree \( n \) of \( F \). Then \( C=F(x) \) is the field of all rational functions with coefficients in \( F \) of a root \( x \) of an equation \( \phi(\omega)=0 \) which has coefficients in \( F \), degree \( n \), and transitive group \( G \) for \( F \).

The problem of the construction of all equations of degree \( n \) and group \( G \) is evidently equivalent to the problem of the construction of all corresponding fields \( C \). Moreover the construction of a set of canonical equations \( \psi(\omega) = 0 \) with the property that every \( C=F(x) \) of degree \( n \) and group \( G \) is equal to an \( F(y) \) defined by a \( \psi(\omega) = 0 \) provides a solution of both problems.

One of the most important problems in the algebraic theory of fields is the construction of all cyclic fields of degree \( n \) over \( F \). This is the case where \( G \) consists of the \( n \) distinct powers \( S^i \) (\( i = 0 \), \( 1,\dots, n-1 \)) of a single substitution \( S \). In this case \( G \) is also the group of all automorphisms of \( C \). Moreover this problem has been reduced to the case \( n = p^e \), \( p \) a prime.

Cyclic fields of degree 2, \( 2^2 \) have been constructed. In the present paper we shall use purely algebraic methods to construct all cyclic fields of degree \( 2^3 = 8 \) [Mertens 1916] over any non-modular field \( F \).

[58]A. A. Albert: “A note on the Dickson theorem on universal ternaries,” Bull. Am. Math. Soc. 39 : 8 (1933), pp. 585–588. MR 1562680 JFM 59.0181.02 Zbl 0007.33704 article

A form \( f \) with integer coefficients in integer variables is called universal if it represents all positive and negative integers. Evidently, since \( f \) is homogeneous, it represents zero for the variables all zero. In case \( f = 0 \) for integral values of the variables not all zero \( f \) is called a zero form.

L. E. Dickson [1930] has given a number-theoretic proof of his theorem that every universal ternary quadratic form is a zero form. But his proof is highly technical and consequently quite long and complicated. In the present note I shall give an almost trivial rational proof of Dickson’s result. I shall also prove a generalization of his theorem for ternaries over any nonmodular field \( F \).

[59]A. A. Albert: “Non-cyclic algebras of degree and exponent four,” Trans. Am. Math. Soc. 35 : 1 (1933), pp. 112–121. MR 1501674 JFM 59.0158.02 Zbl 0006.15103 article

I have recently [1932] proved the existence of non-cyclic normal division algebras. The algebras I constructed are algebras \( A \) of order sixteen (degree four, so that every quantity of \( A \) is contained in some quartic sub-field of \( A \)) containing no cyclic quartic sub-field and hence not of the cyclic (Dickson) type. But each \( A \) is expressible as a direct product of two (cyclic) algebras of degree two (order four). Hence the question of the existence of non-cyclic algebras not direct products of cyclic algebras, and therefore of essentially more complex structures than cyclic algebras, has remained unanswered.

The exponent of a normal division algebra \( A \) is the least integer \( e \) such that \( A^e \) is a total matric algebra. A normal division algebra of degree four has exponent two or four according as it is or is not expressible as a direct product of algebras of degree two. I shall prove here that there exist non-cyclic normal division algebras of degree and exponent four, algebras of a more complex structure than any previously constructed normal division algebras.

[60]A. A. Albert: “On universal sets of positive ternary quadratic forms,” Ann. Math. (2) 34 : 4 (October 1933), pp. 875–878. MR 1503139 JFM 59.0181.04 Zbl 0007.39204 article

[61]A. A. Albert: “On primary normal division algebras of degree eight,” Bull. Am. Math. Soc. 39 : 4 (1933), pp. 265–272. An abstract was published in Bull. Am. Math. Soc. 39:1 (1933). MR 1562597 JFM 59.0159.01 Zbl 0007.05503 article

A normal division algebra \( A \) of degree \( n \) over \( F \) will be called primary if \( A \) is not expressible as a direct product of two normal division algebras \( B \) and \( C \), where neither \( B \) nor \( C \) has degree unity. It is well known that necessarily \( n = p^e \), \( p \) a prime, if \( A \) is primary. Moreover, if \( n = p^e \), then a sufficient condition that \( A \) be primary is that \( A \) shall have exponent \( n \).

I have recently proved [1932] that if \( A \) has degree four then \( A \) is primary if and only if \( A \) has exponent four. In the present paper I shall prove that there exist primary (cyclic) normal division algebras of degree eight but exponent four so that the above sufficient condition is actually not necessary.

[62]A. A. Albert: “A note on the equivalence of algebras of degree two,” Bull. Am. Math. Soc. 39 : 4 (1933), pp. 257–258. An abstract was published in Bull. Am. Math. Soc. 39:3 (1933). MR 1562594 JFM 59.0158.04 Zbl 0007.05502 article

The simplest type of normal simple algebra over any nonmodular field \( F \) is the cyclic algebra of degree two (order four) \begin{gather*} Q(\alpha,\beta) = (1,i,j,ij),\\ ji = -ij, \quad i^2 = \alpha \neq 0, \quad j^2 = \beta \neq 0, \end{gather*} (\( \alpha \) and \( \beta \) in \( F \)), the so-called generalized quaternion algebra over \( F \). Of great importance in the theory of linear algebras is the problem of finding conditions that two given normal simple algebras of the same degree shall be equivalent. But this problem has not, as yet, been explicitly solved even for the above simplest case of algebras of degree two except when \( F \) is an algebraic field [Albert 1930; Hasse 1932]. The purpose of this brief note is to give a simplification of my own previous results for rational algebras of degree two and thereby to give simple explicit conditions that any two generalized quaternion algebras over any non-modular field \( F \) shall be equivalent.

[63]A. A. Albert: “On certain imprimitive fields of degree \( p^2 \) over \( P \) of characteristic \( p \),” Bull. Am. Math. Soc. 39 : 11 (1933), pp. 870. Abstract only; paper in Ann. Math. 35:2 (1934). JFM 59.0168.20 article

[64]A. A. Albert: “On universal sets of positive ternary quadratic forms,” Bull. Am. Math. Soc. 39 : 5 (1933), pp. 345–346. Abstract only; paper in Ann. Math. 34:4 (1933). JFM 59.0187.07 article

[65]A. A. Albert: “Integral domains of rational generalized quaternion algebra,” Bull. Am. Math. Soc. 39 : 7 (1933), pp. 501. Abstract only; paper in Bull. Am. Math. Soc. 40:2 (1934). JFM 59.0168.18 article

[66]A. A. Albert: “Normal division algebras over a modular field,” Bull. Am. Math. Soc. 39 : 11 (1933), pp. 870–871. Abstract only; paper in Trans. Am. Math. Soc. 36:2 (1934). JFM 59.0168.22 article

[67]A. A. Albert: “On primary normal division algebras of degree eight,” Bull. Am. Math. Soc. 39 : 1 (1933), pp. 41. Abstract only; paper in Bull. Am. Math. Soc. 39:4 (1933). JFM 59.0168.15 article

[68]A. A. Albert: “Cyclic fields of degree eight,” Bull. Am. Math. Soc. 39 : 1 (1933), pp. 41. Abstract only; paper in Ann. Math. 35:4 (1933). JFM 59.0168.16 article

[69]A. A. Albert: “A note on the equivalence of algebras of degree two,” Bull. Am. Math. Soc. 39 : 3 (1933), pp. 198–199. Abstract only; paper in Bull. Am. Math. Soc. 39:4 (1933). JFM 59.0168.17 article

[70]A. A. Albert: “Normal division algebras over algebraic number fields not of finite degree,” Bull. Am. Math. Soc. 39 : 9 (1933), pp. 667. Abstract only; paper in Bull. Am. Math. Soc. 39:10 (1933). JFM 59.0168.19 article

[71]A. A. Albert: “Cyclic fields of degree \( p^n \) over \( F \) of characteristic \( p \),” Bull. Am. Math. Soc. 40 : 8 (1934), pp. 625–631. An abstract was published in Bull. Am. Math. Soc. 40:3 (1934). MR 1562919 Zbl 0010.00402 article

[72]A. A. Albert: “Normal division algebras over a modular field,” Trans. Am. Math. Soc. 36 : 2 (1934), pp. 388–394. An abstract was published in Bull. Am. Math. Soc. 39:11 (1933). MR 1501750 JFM 60.0105.02 Zbl 0009.19502 article

An infinite field \( F \) is called perfect if either \( F \) is non-modular or every quantity of \( F \) has the form \( \beta^p \) where \( p \) is the characteristic of \( F \) and \( \beta \) is in \( F \). In any consideration of normal division algebras \( D \) over \( F \) the property that \( F \) is perfect is used only when we consider quantities of \( D \) and the minimum equations of these quantities. But if the degree \( n \) of \( D \) is not divisible by the characteristic \( p \) of \( F \), then the assumption that \( F \) is perfect evidently has no value and is a needless extremely strong restriction on \( F \).

In most of the papers on the structure of normal division algebras written recently in Germany, the assumption has been that \( F \) is perfect. But I shall prove here that if \( F \) is perfect of characteristic \( p \), then \( n \) is not divisible by \( p \). Hence it is now necessary to consider algebras of degree \( p^e \) over \( F \) of characteristic \( p \), where \( F \) is not perfect.

I shall give here a brief discussion of the validity of the major results on algebras over non-modular fields when \( F \) is assumed to be merely any infinite field. Moreover, I shall determine all normal division algebras of degree two over \( F \) of characteristic two, of degree three over \( F \) of characteristic three.

[73]A. A. Albert: “A solution of the principal problem in the theory of Riemann matrices,” Ann. Math. (2) 35 : 3 (July 1934), pp. 500–515. An abstract was published in Bull. Am. Math. Soc. 40:3. MR 1503176 Zbl 0010.00401 article

The principal problem in the theory of Riemann matrices is that of determining the multiplication algebra \( D \) of any pure Riemann matrix \( \omega \). This problem is of great importance in the transcendental theory of algebraic geometry, and its solution provides a solution of many related algebraic geometric questions from which it arose.

Algebra \( D \) is a normal division algebra of degree \( n \) (order \( n^2 \)) over its centrum \( K \). The centrum \( K \) is an algebraic field \( R(\rho) \) of all rational functions with rational coefficients of an abstract root \( \rho \) of an irreducible rational equation \( f(r) = 0 \). We call \( \omega \) a Riemann matrix of the first or second kind according as \( f(r) = 0 \) does or does not have all real roots.

If \( \omega \) is of the first kind, then [Albert 1932] \( n=1,2 \); I have determined [1934] the resulting algebras \( D \), and have proved [1934] that there exist pure Riemann matrices \( \omega \) with any one of the so determined algebras \( D \) as mutiplication algebra. There remains the case of matrices \( \omega \) of the second kind.

The remaining more difficult problem will be solved in the present paper. I shall first define a type of algebra \( \mathfrak{A} \), shall prove the existence of Riemann matrices \( \omega \) with \( \mathfrak{A} \) as multiplication algebra, and shall finally show that the multiplication algebra \( D \) of any pure Riemann matrix \( \omega \) of the second kind is an algebra \( \mathfrak{A} \).

[74]A. A. Albert: “Normal division algebras of degree 4 over \( F \) of characteristic 2,” Am. J. Math. 56 : 1–4 (1934), pp. 75–86. MR 1507931 JFM 60.0105.01 Zbl 0008.24202 article

I have determined [1932a] all normal division algebras of degree 4 over any non-modular field \( F \) and it is evident that the determination is valid when \( F \) is any infinite modular field of characteristic \( p\neq 2 \). There remains the case \( p=2 \).

In the non-modular case there are but two types of algebras, the cyclic algebras and a second type containing algebras which may or may not be cyclic. I have proved [1932b; 1933] the existence of both primary and non-primary non-cyclic algebras over a non-modular \( F \).

In the present paper I shall assume that \( F \) is any infinite field of characteristic \( p=2 \) and shall determine all normal division algebras \( D \) of degree 4 (order 16) over \( F \). As in the non-modular case the above two types of algebras appear. However I shall prove that every non-primary \( D \) is cyclic. I shall also prove that a necessary and sufficient condition that \( D \) be cyclic is that \( D \) contain a quantity \( t \) not in \( F \) such that \( t^2 \) is in \( F \).

Finally I shall give a construction of the non-cyclic type of algebra \( D \) and shall prove that \( D \) is a non-cyclic primary normal division algebra if and only if a certain quadratic form in nine variables (and with coefficients determined by the multiplication table of \( D \)) is not a zero form.

[75]A. A. Albert: “Integral domains of rational generalized quaternion algebras,” Bull. Am. Math. Soc. 40 : 2 (1934), pp. 164–176. An abstract was published in Bull. Am. Math. Soc. 39:7 (1933). MR 1562813 Zbl 0008.29301 article

[76]A. A. Albert: “On normal Kummer fields over a non-modular field,” Trans. Am. Math. Soc. 36 : 4 (1934), pp. 885–892. MR 1501774 JFM 60.0915.03 Zbl 0010.14902 article

Let \( F \) be any non-modular field, \( p \) an odd prime, \( \zeta \neq 1 \) a \( p \)th root of unity. Suppose that \( \mu \) in \( F(\zeta) \) is not the \( p- \)th power of any quantity of \( F(\zeta) \) so that the equation \( y^p=\mu \), is irreducible in \( F(\zeta) \). Then the field \( F(y,\zeta) \) is called a Kummer field over \( F \).

In the present paper we shall give a formal construction of all normal Kummer fields over \( F \). This is equivalent to a construction of all fields \( F(x) \) of degree \( p \) over \( F \) such that \( F(x,\zeta) \) is cyclic of degree \( p \) over \( F(\zeta) \). In particular we provide a construction of all cyclic fields of degree \( p \) over \( F \).

We shall also apply the cyclic case to prove that a normal division algebra \( D \) of degree \( p \) over \( F \) is cyclic if and only if \( D \) contains a quantity \( y \) not in \( F \) such that \( y^p=\gamma \) in \( F \).

[77]A. A. Albert: “On certain imprimitive fields of degree \( p^2 \) over \( P \) of characteristic \( p \),” Ann. Math. (2) 35 : 2 (April 1934), pp. 211–219. An abstract was published in Bull. Am. Math. Soc. 39:11 (1933). MR 1503156 JFM 60.0915.02 Zbl 0009.24302 article

[78]A. A. Albert: “The principal matrices of a Riemann matrix,” Bull. Am. Math. Soc. 40 : 12 (1934), pp. 843–846. An abstract was published in Bull. Am. Math. Soc. 40:9 (1934). MR 1562985 JFM 60.0908.01 Zbl 0010.29202 article

[79]A. A. Albert: “On a certain algebra of quantum mechanics,” Ann. Math. (2) 35 : 1 (January 1934), pp. 65–73. MR 1503142 JFM 60.0902.03 Zbl 0008.42104 article

P. Jordan, J. von Neumann, and E. Wigner [1934] have discussed certain linear real non-associative algebras of importance in quantum mechanics. Their algebras \( \mathfrak{M} \) satisfy the ordinary postulates for addition, the commutative law for multiplication, and the distributive law, but they are non-associative.

In the paper quoted above it is shown that, with a single exception, every algebra satisfying the above postulates is equivalent to an algebra \( \mathfrak{M} \) whose elements are ordinary real matrices \( x, y,\dots \) with products \( xy \) in \( \mathfrak{M} \) defined by quasi-multiplication, \[ xy = \tfrac{1}{2}(x\cdot y + y\cdot x), \] where \( x\cdot y \) is the ordinary matrix product. This single exception is the algebra \( \mathfrak{M}_3^8 \) of all three rowed Hermitian matrices with elements in the real non-associative algebra \( C \) of Cayley numbers.

The algebras obtained by quasi-multiplication of real matrices were considered in earlier papers so that, as stated by the above authors, this seemingly exceptional case, if proved really exceptional, is the only algebra of the above type which could lead to any new form of quantum mechanics.

In the present paper I shall prove that \( \mathfrak{M}_3^8 \) is a new algebra and that it is not equivalent to any algebra obtained by quasi-multiplication of real matrices. Moreover, I shall show that the relation \[ x(yx^2) = (xy)x^2 \] is satisfied by \( \mathfrak{M}_3^8 \).

[80]A. A. Albert: “On the construction of Riemann matrices. I,” Ann. Math. (2) 35 : 1 (January 1934), pp. 1–28. MR 1503140 JFM 60.0908.02 Zbl 0010.00304 article

[81]A. A. Albert: “Involutorial simple algebras and real Riemann matrices,” Proc. Natl. Acad. Sci. U.S.A. 20 : 12 (1934), pp. 676–681. An abstract was published in Bull. Am. Math. Soc. 40:11 (1934). See also Ann. Math. (2) 36:4 (1935). JFM 60.0909.01 Zbl 0010.29203 article

A simple algebra \( A \) over \( F \) is said to be \( J \)-involutorial if there is a self correspondence \( J \) of \( A \) such that \[ (a+b)^J= a^J+b^J, \quad (ab)^J = b^J a^J, \quad (a^J)^J = a, \quad \lambda^J = \lambda \] for every \( a \) and \( b \) of \( A \) and \( \lambda \) of \( F \). The structure problem for \( J \)-involutorial simple algebras is closely connected with the study of the multiplication algebras of real Riemann matrices. The author has completed a study of both of these problems and also has shown how closely related they are to Weyl’s recent generalization of Riemann matrices. A complete reduction of any impure real Riemann matrix to pure components has also been obtained.

[82]A. A. Albert: “Cyclic fields of degree \( p^n \) over \( F \) of characteristic \( p \),” Bull. Am. Math. Soc. 40 : 3 (1934), pp. 218. Abstract only; paper in Bull. Am. Math. Soc. 40:8 (1934). JFM 60.0113.22 article

[83]A. A. Albert: “The principal matrices of a Riemann matrix,” Bull. Am. Math. Soc. 40 : 9 (1934), pp. 652. Abstract only; article in Bull. Am. Math. Soc. 40:12 (1934). JFM 60.0059.05 article

[84]A. A. Albert: “A solution of the principal problem in the theory of Riemann matrices,” Bull. Am. Math. Soc. 40 : 3 (1934), pp. 217. Abstract only; paper in Ann. Math. 35:3 (1934). JFM 60.0916.11 article

[85]A. A. Albert: “Involutorial simple algebras and real Riemann matrices,” Bull. Am. Math. Soc. 40 (1934), pp. 808. Abstract only; paper in Proc. Natl. Acad. Sci. U.S.A. 20:12 (1934). JFM 60.0916.10 article

[86]A. A. Albert: “On the construction of Riemann matrices. II,” Ann. Math. (2) 36 : 2 (April 1935), pp. 376–394. MR 1503230 JFM 61.1037.02 Zbl 0011.38904 article

[87]A. A. Albert: “A note on the Poincaré theorem on impure Riemann matrices,” Ann. Math. (2) 36 : 1 (January 1935), pp. 151–156. MR 1503214 JFM 61.0124.01 Zbl 0011.00603 article

A \( p \) by \( 2p \) complex matrix \( \omega \) is a Riemann matrix if there exists a rational \( 2p \)-rowed skew-symmetric matrix \( C \) such that \( \omega C\omega^{\prime} = 0 \), \( i\omega C\overline{\omega}^{\prime} \) is positive definite. If \( \alpha \) is \( p \)-rowed complex and non-singular, \( A \) is \( 2p \)-rowed rational and non-singular, then \( \alpha\omega A \) and \( \omega \) are said to be isomorphic.

The period matrix of the abelian integrals of the first kind on a Riemann surface is a Riemann matrix. It was in this connection that Poincar/’e proved that if \( \omega \) is impure, that is \( \omega \) is isomorphic to \[ \left( \begin{array}{cc} \omega_1 & 0\\ \omega_3 & \omega_2 \end{array} \right) ,\] where \( \omega_1 \) is a Riemann matrix, then \( \omega \) is also isomorphic to \[ \left( \begin{array}{cc} \omega_1 & 0\\ 0 & \omega_2 \end{array} \right) \] where \( \omega_2 \) is also a Riemann matrix.

The above theorem was generalized to the case where \( \omega \) is any Riemann matrix by G. Scorza. Scorza used a rather complicated projective geometric method and his proof (as well as all previous proofs) is not very simple. I shall give here an elementary matrix proof of the above theorem and the later Scorza reduction of any impure Riemann matrix to its pure components.

[88]A. A. Albert: “On cyclic fields,” Trans. Am. Math. Soc. 37 : 3 (1935), pp. 454–462. MR 1501797 JFM 61.0124.02 Zbl 0011.29005 article

The most interesting algebraic extensions of an arbitrary field \( F \) are the cyclic extension fields \( Z \) of degree \( n \) over \( F \). I have recently given constructions of such fields for the case \( n = p \) [1934a], a prime, when the characteristic of \( F \) is not \( p \), and for the case \( n = p^e \) [1934b] when the characteristic of \( F \) is \( p \). Moreover it is well known that when \( F \) contains all the \( n \)th roots of unity then \( Z = F(x) \), \( x^n = \alpha \) in \( F \).

The last result above does not provide a construction of all cyclic fields \( Z \) over \( F \) since in general \( F \) does not contain these \( n \)th roots. Moreover if we adjoin these roots to \( F \) and so extend \( F \) to a field \( K \) the composite \( (Z,K) \) over \( K \) may not have degree \( n \). Finally even if \( (Z,K) \) over \( K \) does have degree \( n \) then it is necessary to give conditions that a given field \( K(x) \), \( x^n=\alpha \) in \( F \), shall have the form \( (Z,K) \) with \( Z \) cyclic over \( F \). This has not been done and is certainly not as simple as the considerations I shall make here.

It is well known that if \( n = p_1^{e_1}\cdots p_t^{e_t} \) with \( p_i \) distinct primes, then \( Z \) is the direct product \[ Z = Z_1 \times \cdots \times Z_t \] where \( Z_i \) is cyclic of degree \( p \) over \( F \). Hence it suffices to consider the case \( n = p^e \), \( p \) a prime. I have already done so [Albert 1934b] for the case where \( F \) has characteristic \( p \). In the present paper I shall make analogous considerations for the case where \( F \) has characteristic not \( p \) by first studying the case where \( F \) contains a primitive \( p \)th root of unity \( \zeta \) and later giving complete conditions for the case where \( F \) does not contain \( \zeta \).

[89]A. A. Albert: “Involutorial simple algebras and real Riemann matrices,” Ann. Math. (2) 36 : 4 (October 1935), pp. 886–964. See also Proc. Natl. Acad. Sci. U.S.A. 20:12 (1934). MR 1503260 JFM 61.1038.01 Zbl 0012.39102 article

Any algebraic theory depends in part on the properties of its reference field \( \mathfrak{K} \). In particular it is generally desirable to know whether or not the Hilbert Irreducibility theorem holds in \( \mathfrak{K} \).

W. Franz studied [1931] fields with this question in mind and proved that \( \mathfrak{K} \) is a Hilbert Irreducibility field if it is a separable algebraic extension of finite degree over a Hilbert Irreducibility field \( \mathfrak{F} \). This is however insufficient for the theory of algebras over a modular field since the field \( \mathfrak{K} \) may be inseparable over \( \mathfrak{F} \).

We shall treat this latter case here and shall show that any algebraic extension \( \mathfrak{K} \) of finite degree over a Hilbert Irreducibility field \( \mathfrak{F} \) is a Hilbert irreducibility field. Moreover this result will be a sufficiently good tool for the researches of later chapters.

[90]A. A. Albert: “Simple algebras of degree \( p^e \) over a centrum of characteristic \( p \),” Bull. Am. Math. Soc. 41 : 7 (1935), pp. 489. Abstract only; paper in Trans. Am. Math. Soc. 40:1 (1936). JFM 61.0127.09 article

[91]A. A. Albert: “Normal division algebras of degree \( p^e \) over \( F \) of characteristic \( p \),” Trans. Am. Math. Soc. 39 : 1 (January 1936), pp. 183–188. MR 1501840 JFM 62.0101.01 Zbl 0013.10202 article

In a recent paper [1934] I proved that a normal division algebra \( D \) of degree \( p \), a prime, over a field \( F \) of characteristic not \( p \), is cyclic if and only if \( D \) contains a sub-field \( F(y) \), \( y^p = \gamma \) in \( F \). This result evidently leads to the conjecture that any normal division algebra \( D \) of degree \( n \) over \( F \) is cyclic over \( F \) if and only if \( D \) contains a maximal sub-field, \( F(y) \), \( y^n = \gamma \) in \( F \).

The conjectured criterion given above would be of fundamental importance for the theory of the structure of normal division algebras. Without loss of generality we may assume that \( n=p^e \), \( p \) a prime, and the theory then gives rise to two distinct cases according as \( F \) does or does not have characteristic \( p \). We shall consider the former case here and give a brief simple proof of the criterion.

[92]A. A. Albert: “Simple algebras of degree \( p^e \) over a centrum of characteristic \( p \),” Trans. Am. Math. Soc. 40 : 1 (July 1936), pp. 112–126. An abstract was published in Bull. Am. Math. Soc. 41:7 (1935). MR 1501866 JFM 62.0101.02 Zbl 0014.29102 article

The study of normal simple algebras \( A \) over a field \( F \) has been reduced to the case where the degree of \( A \) is a power of a prime \( p \). The theory then splits sharply into two cases distinguished by the hypothesis that the characteristic of \( F \) is or is not \( p \). We shall restrict our attention to the former case.

It is well known that every normal simple algebra \( A \) over \( F \) is similar to a crossed product \( B \). But this result is of little aid in a study of \( A \), since in fact the degree of \( B \) is in general not a power of \( p \). We shall prove here, however, that in our case every \( A \) is similar to a cyclic algebra whose degree is a power of \( p \).

If \( K \) is obtained from \( F \) by adjoining the \( p^e \)th roots of quantities of \( F \) to \( F \), for fixed \( e \), then \( K \) is said to have exponent \( e \) over \( F \). We shall show that the exponent of an algebra \( A \) is \( p^e \) where \( e \) is the exponent of the above \( K \) of least exponent, which splits \( A \). Moreover \( A \) has exponent \( p^e \) only if \( A \) is similar to a direct product of cyclic division algebras \( D_i \) whose exponents and degrees are equal to \( p^{e_i}\leq p^e \), \( D_1 \) of degree \( p^e \).

[93]A. A. Albert: “Normalized integral bases of algebraic number fields, I,” Bull. Am. Math. Soc. 42 : 11 (1936), pp. 811. Abstract only; paper in Ann. Math. 38:4 (1937). JFM 62.0172.06 article

[94]A. A. Albert: “\( p \)-algebras over a field generated by one indeterminate,” Bull. Am. Math. Soc. 43 : 10 (1937), pp. 733–736. MR 1563627 JFM 63.0875.01 Zbl 0017.24501 article

The structure of all division algebras over the simplest type of non-modular field, the field of all rational numbers, has been determined [Hasse and Albert 1932]. The correspondingly simplest type of infinite modular field is the simple transcendental extension \( K = F(x) \) of a finite field \( F \). Every division algebra \( D \) over such a \( K \) is a normal division algebra of degree \( n \) over a centrum \( G \) which is algebraic of finite degree over \( K \). It is well known that the problem of determining the structure of \( D \) is reducible to the case where \( n \) is a power of a prime \( p \). When \( p \) is the characteristic of \( F \) the algebra \( D \) is called a \( p \)-algebra and we shall solve the problem in this case. Our results will be valid if we replace the finite field \( F \) by any perfect field of characteristic \( p \).

The theorem we shall obtain is remarkable not merely because of the character of the result thus derived but also because of the extremely elementary nature of the proof. By using a simple property of the field \( G \) described above we shall show that every \( p \)-algebra with centrum \( G \) is cyclic and of exponent equal to its degree. Moreover this result is due to the unusual fact that all cyclic algebras over \( G \) of the same degree \( p^e \) have a common pure inseparable splitting field.

[95]A. A. Albert: “Normalized integral bases of algebraic number fields. I,” Ann. Math. (2) 38 : 4 (October 1937), pp. 923–957. An abstract was published in Bull. Am. Math. Soc. 42:11 (1936). MR 1503382 JFM 63.0910.01 Zbl 0018.05102 article

Every algebraic number field \( \mathfrak{F} \) of degree \( n \) over the rational number field is defined by a root \( \theta \) of an irreducible rational equation \( f(x) = 0 \) of degree \( n \). One of the most interesting problems in the theory of such fields is that of finding explicit formulae for the quantities of a basis of the integers of \( \mathfrak{F} \) in terms of \( \theta \) and the coefficients of \( f(x) \). This problem has had no general solution and has not been completely solved even for the case \( n=4 \).

Some simplifications in the above formulae may be obtained by choosing \( \theta \) to be an algebraic integer and such that \( a\theta \) is an algebraic integer for rational \( a \) if and only if \( a \) is integral. These are partial normalizations on \( \theta \). Additional simplifications of the formulae are obtained when we choose \( \theta \) to have zero trace, that is the coefficient of \( x^{n-1} \) in \( f(x) \) to be zero.

In the present paper we shall obtain simplifications by making further normalizations on \( \theta \). We shall in fact show that every \( \mathfrak{F} \) of degree \( n > 2 \) is generated by a quantity \( \theta \) such that in addition to the above properties the first two elements of an integral basis of \( \mathfrak{F} \) are \[ 1,\quad \theta \] This result will be completed here by our giving necessary and sufficient conditions on \( f(x) \) that \( \theta \) have the desired properties.

[96]A. A. Albert: “A note on matrices defining total real fields,” Bull. Am. Math. Soc. 43 : 4 (1937), pp. 242–244. MR 1563515 JFM 63.0142.05 Zbl 0016.15003 article

[97]A. A. Albert: Modern higher algebra. University of Chicago Press, 1937. JFM 63.0868.01 Zbl 0017.29201 book

[98]A. A. Albert: “A note on normal division algebras of prime degree,” Bull. Am. Math. Soc. 44 : 10 (1938), pp. 649–652. MR 1563842 JFM 64.0081.04 Zbl 0019.29002 article

[99]A. A. Albert: “On cyclic algebras,” Ann. Math. (2) 39 : 3 (July 1938), pp. 669–682. MR 1503431 JFM 64.0082.02 Zbl 0019.24601 article

[100]A. A. Albert: “Non-cyclic algebras with pure maximal subfields,” Bull. Am. Math. Soc. 44 : 8 (1938), pp. 576–579. MR 1563796 JFM 64.0082.01 Zbl 0019.19302 article

[101]A. A. Albert: “Symmetric and alternate matrices in an arbitrary field. I,” Trans. Am. Math. Soc. 43 : 3 (May 1938), pp. 386–436. MR 1501952 Zbl 0018.34202 article

[102]A. A. Albert: “A quadratic form problem in the calculus of variations,” Bull. Am. Math. Soc. 44 : 4 (1938), pp. 250–253. MR 1563719 JFM 64.0515.01 Zbl 0018.24201 article

[103]A. A. Albert: “The Chicago conference and seminar on algebra,” Bull. Am. Math. Soc. 44 : 11 (1938), pp. 756–757. MR 1563863 article

[104]A. A. Albert: “Quadratic null forms over a function field,” Ann. Math. (2) 39 : 2 (April 1938), pp. 494–505. MR 1503420 JFM 64.0083.01 Zbl 0019.00207 article

[105]A. A. Albert: Modern higher algebra. Cambridge University Press, 1938. Zbl 0019.14705 book

[106]A. A. Albert: Structure of algebras. AMS Colloquium Publications 24. American Mathematical Society (Providence, RI), 1939. MR 0000595 Zbl 0023.19901 book

[107]A. A. Albert: “On ordered algebras,” Bull. Am. Math. Soc. 46 (1940), pp. 521–522. MR 0001972 JFM 66.0114.01 Zbl 0061.05502 article

[108]A. A. Albert: “On \( p \)-adic fields and rational division algebras,” Ann. Math. (2) 41 : 3 (July 1940), pp. 674–693. MR 0002861 JFM 66.0116.01 Zbl 0024.14602 article

We shall use a very simple lemma of what may be called an arithmetical nature on \( p \)-adic fields and shall give a consequent new algebraic discussion of fields \( \mathfrak{K} \) of finite degree and ramification order \( e \) prime to \( p \) over a \( p \)-adic field \( \mathfrak{F} \). We shall construct all such fields, show that the number of such fields inequivalent over \( \mathfrak{F} \) is an explicitly determined divisor of \( e \), and shall determine necessary and sufficient conditions that \( \mathfrak{K} \) be normal over \( \mathfrak{F} \). We shall determine the automorphism group of \( \mathfrak{K} \) over \( \mathfrak{F} \), and prove the rather amazing result that the existence of any abelian field of the given ramification order \( e \) over \( \mathfrak{F} \) implies that al fields with the same \( e \) are abelian over \( \mathfrak{F} \). Finally we shall determine explicitly which abelian fields are cyclic, how many cyclic fields exist which are inequivalent over \( \mathfrak{F} \), the direct factorization into two cyclic fields of all non-cyclic abelian fields.

[109]A. A. Albert: “A rule for computing the inverse of a matrix,” Am. Math. Monthly 48 : 3 (March 1941), pp. 198–199. MR 0003603 JFM 67.0052.03 Zbl 0060.03301 article

[110]A. A. Albert: “Book review: The theory of group characters,” Bull. Am. Math. Soc. 47 : 5 (1941), pp. 357–359. Book by Dudley E. Littlewood (Oxford University Press, 1940). MR 1564250 article

[111]A. A. Albert: Introduction to algebraic theories. University of Chicago Press, 1941. MR 0003590 JFM 67.0045.03 Zbl 0027.00601 book

[112]A. A. Albert: “Division algebras over a function field,” Duke Math. J. 8 : 4 (1941), pp. 750–762. MR 0006168 Zbl 0061.05503 article

The most important splitting fields \( K \) of a division algebra \( D \) of degree \( m \) over its centrum \( F \) are those fields of least possible degree over \( F \). This degree is \( m \) and it is known [Albert 1939, Theorem 4.27] also that all such fields are equivalent to subields of \( D \) of maximal degree over \( F \). It is natural then to propose the question as to the universal existence of a constant splitting field over \( F \) of degree \( m \) over \( L \) for any \( D \). In particular a cyclic field of this kind would provide an extremely simple generation of \( D \).

We shall answer the question just proposed in the negative and shall prove indeed that the least degree \( n \) of a constant splitting field may be arbitrarily large. We shall prove in fact the:

Let \( x \) be an indeterminate over a finite field \( L \), \( F = L(x) \), \( m \) and \( n \) be positive integers. Then there exist division algebras \( D \) of degree \( m \) over \( F \) as centrum and with \( n \) as the least degree over \( L \) of a constant splitting field over \( F \) of \( D \) if and only if \( m \) divides \( n \) and every prime factor of \( n \) divides \( m \).

[113]A. A. Albert: “Quadratic forms permitting composition,” Ann. Math. (2) 43 : 1 (January 1942), pp. 161–177. MR 0006140 JFM 68.0051.02 Zbl 0060.04003 article

[114]A. A. Albert: “Non-associative algebras. II: New simple algebras,” Ann. Math. (2) 43 : 4 (October 1942), pp. 708–723. MR 0007748 Zbl 0061.04901 article

[115]A. A. Albert: “Non-associative algebras. I: Fundamental concepts and isotopy,” Ann. Math. (2) 43 : 4 (October 1942), pp. 685–707. MR 0007747 Zbl 0061.04807 article

[116]A. A. Albert: “The radical of a non-associative algebra,” Bull. Am. Math. Soc. 48 (1942), pp. 891–897. MR 0007396 Zbl 0061.05001 article

Every algebra \( \mathfrak{A} \) which is homomorphic to a semi-simple algebra has an ideal \( \mathfrak{R} \). which we shall call the radical of \( \mathfrak{A} \), such that \( \mathfrak{A} - \mathfrak{R} \) is semi-simple, \( \mathfrak{R} \) is contained in every ideal \( \mathfrak{B} \) of \( \mathfrak{A} \) for which \( \mathfrak{A} - \mathfrak{B} \) is semi-simple.

An algebra \( \mathfrak{A} \) is homomorphic to a semi-simple algebra if and only if \( \mathfrak{A} - \mathfrak{A}\mathfrak{H} \) is not a zero algebra.

If \( \mathfrak{A} - \mathfrak{A}\mathfrak{H} \) is a zero algebra and \( \mathfrak{B} \) is an ideal of \( \mathfrak{A} \) the algebra \( \mathfrak{A} - \mathfrak{B} \) is a zero algebra if and only if \( \mathfrak{B} \) contains \( \mathfrak{A}\mathfrak{H} \).

Let \( \mathfrak{A} \) be homomorphic to a semi-simple algebra. Then either \( \mathfrak{A} - \mathfrak{A}\mathfrak{H} \) is semi-simple and \( \mathfrak{A}\mathfrak{H} \) is the radical of \( \mathfrak{A} \) or \( \mathfrak{A} - \mathfrak{A}\mathfrak{H} \) is the direct sum of a semi-simple algebra and a zero algebra \( \mathfrak{R}_0 = \mathfrak{R} - \mathfrak{A}\mathfrak{H} \) such that \( \mathfrak{R} \) is the radical of \( \mathfrak{A} \).

[117]A. A. Albert: “An inductive proof of Descartes’ rule of signs,” Am. Math. Monthly 50 : 3 (March 1943), pp. 178–180. MR 0007806 Zbl 0060.05004 article

[118]A. A. Albert: “A suggestion for a simplified trigonometry,” Am. Math. Monthly 50 : 4 (1943), pp. 251–253. MR 1525668 article

[119]A. A. Albert: “Quasigroups. I,” Trans. Am. Math. Soc. 54 (1943), pp. 507–519. MR 0009962 Zbl 0063.00039 article

[120]A. A. Albert: “The minimum rank of a correlation matrix,” Proc. Nat. Acad. Sci. U. S. A. 30 : 6 (June 1944), pp. 144–146. MR 0010351 Zbl 0063.00041 article

[121]A. A. Albert: “Quasigroups. II,” Trans. Am. Math. Soc. 55 : 3 (May 1944), pp. 401–419. MR 0010597 Zbl 0063.00042 article

In the first part of this paper [1943] we associated every quasigroup \( \mathfrak{G} \) with the transformation group \( \mathfrak{G}_r \) generated by the right and left multiplications of \( \mathfrak{G} \). We defined isotopy and showed that every quasigroup is isotopic to a loop, that is, a quasigroup \( \mathfrak{G} \) with identity element \( e \). We also defined the concept of normal divisor for all loops and showed that every normal divisor \( \mathfrak{H} \) of \( \mathfrak{G} \) is equal to \( e\Gamma \) where \( \Gamma \) is a normal divisor of \( \mathfrak{G}_r \).

The main purpose of this second part of our paper is that of presenting a proof, using the results above, of the Schreier refinement theorem and the consquent Jordan–Hölder theorem for arbitrary loops. We obtain also a number of special results, among them a construction of all loops \( \mathfrak{G} \) with a given normal divisor \( \mathfrak{H} \) and a given quotient loop \( \mathfrak{G}/\mathfrak{H} \). We use this in the construction of all loops of order six with a subloop, necessarily a normal divisor, of order three. We classify these loops into nonisotopic classes and show also that all quasigroups of order five are isotopic to one of two nonisotopic loops.

[122]A. A. Albert: “The matrices of factor analysis,” Proc. Nat. Acad. Sci. U. S. A. 30 : 4 (April 1944), pp. 90–95. MR 0009829 Zbl 0063.00040 article

[123]A. A. Albert: “Two element generation of a separable algebra,” Bull. Am. Math. Soc. 50 (1944), pp. 786–788. MR 0011080 Zbl 0061.05501 article

The minimum rank of an algebra \( A \) over a field \( F \) is defined to be the least number \( r = r(A) \) of elements \( x_1,\dots,x_r \) such that \( A \) is the set of all polynomials in \( x_1,\dots,x_r \) with coefficients in \( F \). In what follows we shall assume that \( A \) is an associative algebra of finite order over an infinite field \( F \).

It is well known that \( r(A) = 1 \) if \( A \) is a separable field over \( F \) and that \( r(A) = 2 \) if \( A \) is a total matric algebra over \( F \). Over fourteen years ago I obtained but did not publish the result that \( r(A) = 2 \) if \( A \) is a central division algebra over \( F \). The purpose of this note is to provide a brief proof of the generalization which states that if \( A \) is any separable algebra over \( F \) then \( r(A) = 1 \) or 2 according as \( A \) is or is not commutative.

[124]A. A. Albert: “Quasiquaternion algebras,” Ann. Math. (2) 45 : 4 (October 1944), pp. 623–638. MR 0011081 Zbl 0061.05002 article

We shall consider here certain algebras of order four over a general field \( \mathfrak{F} \). They are a generalization of the “generalized quaternion” algebras of L. E. Dickson which have proved to be such an important special case in the associative theory, and speicalize to the associative case when a single parameter \( \tau = 0 \). The algebras are not associative if and only if \( \tau \neq 0,1 \) and are central simple.

We shall show that the theory of isomorphism, automorphisms, proper subalgebras, and division algebras for these algebras is actually much simpler than in the associative case, a result quite contrary to what might have been expected. Indeed, the theory is remarkably like the theory of quadratic fields. We shall also obtain a complete study of the conditions that two of our quasiquaternion division algebras over a field of characteristic not two be isotopic.

[125]A. A. Albert: “Algebras derived by non-associative matrix multiplication,” Am. J. Math. 66 : 1 (January 1944), pp. 30–40. MR 0009949 Zbl 0061.05003 article

There are four ways to form an \( n \)-rowed matrix whose determinant is the product of the determinants of two \( n \)-rowed matrices, and only one of these, the row by column product, is an associative product. The row by row, column by column, and column by row products determine algebras which are not associative and it has recently been suggested to the author, in conversation, that these algebras have applications to some problems of physics.

We shall study the structure of such algebras here with particular attention to those algebras obtained by the three non-associative products from associative algebras \( \mathfrak{C} \) such that \( \mathfrak{C} \) contains the transpose \( x^{\prime} \) of every \( x \) of \( \mathfrak{C} \). We shall indeed obtain a general structure theory not merely for such algebras but for the case where \( \mathfrak{C} \) is any algebra with an involution \( J \). Thus our results will include the case where \( J \) is the conjugate transpose operation. We shall also show that there are linear spaces of real matrices closed under row by row (or column by column) multiplication and not under row by column multiplication, but that these row by row algebras cannot be semi-simple.

[126]A. A. Albert: “Book review: Galois theory,” Bull. Am. Math. Soc. 51 : 5 (1945), pp. 359. Book by Emil Artin (University of Notre Dame Press, 1942). MR 1564714 article

[127]A. A. Albert: “On Jordan algebras of linear transformations,” Trans. Am. Math. Soc. 59 (1946), pp. 524–555. MR 0016759 Zbl 0061.05101 article

In the present paper we shall obtain the fundamental structure theorems for Jordan algebras over a nonmodular field. We shall derive analogues for Jordan algebras of the Lie and Engel theorems on solvable Lie algebras and shall obtain a trace criterion for the existence of a nonzero radical. The results imply a Pierce decomposition for a Jordan algebra relative to an idempotent which is almost precisely that known for any associative algebra. Then semisimple Jordan algebras have unity elements and are direct sums of simple algebras. The center of a simple Jordan algebra is a field and every simple Jordan algebra may be expressed as a central simple algebra over its center. If \( \mathfrak{A} \) is central simple every scalar extension \( \mathfrak{A}_{\mathfrak{K}} \) of \( \mathfrak{A} \) is simple.

We shall also show that if \( u \) is any primitive idempotent of a simple algebra \( \mathfrak{A} \) the associative product \( u\mathfrak{A} u \) is a simple subalgebra of \( \mathfrak{A} \) and has \( u \) as unity quantity. We shall call \( \mathfrak{A} \) a reduced algebra if \( \mathfrak{A} \) is simple and \( u\mathfrak{A}u \) has order one for every primitive idempotent \( u \) of \( \mathfrak{A} \). Every central simple algebra has a scalar extension field \( \mathfrak{K} \) such that \( \mathfrak{A}_{\mathfrak{K}} \) is a reduced algebra, and we shall determine all reduced Jordan algebras.

[128]A. A. Albert: College algebra. McGraw-Hill (New York), 1946. Reprinted by Univ. Chicago press in 1963. book

[129]A. A. Albert: “A structure theory for Jordan algebras,” Ann. Math. (2) 48 : 3 (July 1947), pp. 546–567. MR 0021546 Zbl 0029.01003 article

[130]A. A. Albert: “The Wedderburn principal theorem for Jordan algebras,” Ann. Math. (2) 48 : 1 (January 1947), pp. 1–7. MR 0019601 Zbl 0029.01002 article

[131]A. A. Albert: “Absolute valued real algebras,” Ann. Math. (2) 48 : 2 (April 1947), pp. 495–501. MR 0020550 Zbl 0029.01001 article

Let \( \mathfrak{D} \) be an algebra over the field \( \mathfrak{R} \) of all real numbers. Then we call \( \mathfrak{D} \) an absolute valued algebra if there is a function \( \phi(a) \) on \( \mathfrak{D} \) to \( \mathfrak{R} \) such that

\( \phi(0) = 0 \),
\( \phi(a) > 0 \) if \( a \neq 0 \),
\( \phi(ab) = \phi(a)\phi(b) \),
\( \phi(a+b) \leq \phi(a) + \phi(b) \),
\( \phi(\alpha a) = |\alpha|\phi(a) \) for every \( a \) and \( b \) of \( \mathfrak{D} \) and real \( \alpha \) where \( |\alpha| \) is the ordinary absolute value.

Then \( \phi(a\pm b) \geq |\phi(a) - \phi(b)| \). Also \( \mathfrak{D} \) must be a division algebra. For if \( a\neq 0 \), \( b\neq 0 \), then

\( \phi(a) > 0 \),
\( \phi(b) > 0 \),
\( \phi(ab) = \phi(a)\phi(b) > 0 \), \( ab\neq 0 \).

It is well known that the only alternative division algebras over \( \mathfrak{R} \) are the field \( \mathfrak{R} \), the field \( \mathfrak{C} \) of all complex numbers, the algebra \( \mathfrak{Q} \) of all real quaternions, and the eight dimensional algebra \( \mathfrak{B} \) of all real Cayley numbers. Each of these algebras has a conjugate operation \( x\to \bar{x} \) such that \( x\bar{x} \) is a positive real number, and it is easy to see that \( \phi(x) = \sqrt{x\bar{x}} \) defines an absolute value for each algebra. A known result then implies that \( \phi(a) \) is unique. We shall prove here that the alternative division algebras are the only absolute valued real algebras with a unity quantity. This characterizes the Cayley algebra as the only absolute valued real algebra with a unity quantity which is not associative. We shall also show that all absolute valued real algebras without a unity quantity are certain isotopes of \( \mathfrak{Q} \) and \( \mathfrak{B} \).

A real algebra \( \mathfrak{A} \) is called a normed algebra if the function \( \phi(a) \) is assumed to satisfy \[ \phi(ab) \leq \phi(a)\phi(b) \] rather than the equality. We shall show here that every real algebra is a normed algebra. Then there exist normed algebras which are not absolute valued.

[132]A. A. Albert: “Power-associative rings,” Trans. Am. Math. Soc. 64 (1948), pp. 552–593. MR 0027750 Zbl 0033.15402 article

[133]A. A. Albert: “On the power-associativity of rings,” Summa Brasil. Math. 2 : 2 (1948), pp. 21–32. MR 0026044 Zbl 0039.26403 article

[134]A. A. Albert: “On right alternative algebras,” Ann. Math. (2) 50 : 2 (April 1949), pp. 318–328. MR 0028828 Zbl 0033.15501 article

[135]A. A. Albert: “A theory of trace-admissible algebras,” Proc. Nat. Acad. Sci. U. S. A. 35 (1949), pp. 317–322. MR 0030511 Zbl 0033.15403 article

[136]A. A. Albert: “Absolute-valued algebraic algebras,” Bull. Am. Math. Soc. 55 : 8 (1949), pp. 763–768. MR 0030941 Zbl 0033.34901 article

An algebra \( \mathfrak{A} \) over a field \( \mathfrak{F} \) is a vector space over \( \mathfrak{F} \) which is closed with respect to a product \( xy \) which is linear in both \( x \) and \( y \). The product is not necessarily associative. Every element \( x \) of \( \mathfrak{A} \) generates a subalgebra \( \mathfrak{F}[x] \) of \( \mathfrak{A} \) and we call \( \mathfrak{A} \) an algebraic algebra if every \( \mathfrak{F}[x] \) is a finite-dimensional vector space over \( \mathfrak{F} \).

We have shown elsewhere [1947] that every absolute-valued real finite-dimensional algebra has dimension 1, 2, 4, or 8 and is either the field \( \mathfrak{R} \) of all real numbers, the complex field \( \mathfrak{C} \), the real quaternion algebra \( \mathfrak{Q} \), the real Cayley algebra \( \mathfrak{D} \), or certain isotopes without unity quantities of \( \mathfrak{Q} \) and \( \mathfrak{D} \). In the present paper we shall extend these results to algebraic algebras over \( \mathfrak{R} \) showing that every algebraic algebra over \( \mathfrak{R} \) with a unity quantity is finite-dimensional and so is one of the algebras listed above. The results are extended immediately to absolute-valued algebraic division algebras, that is, to algebras without unity quantities whose nonzero quantities form a quasigroup.

[137]A. A. Albert: “A note of correction,” Bull. Am. Math. Soc. 55 (1949), pp. 1191. Correction to paper in Bull. Am. Math. Soc. 55:8 (1949). MR 0030942 article

[138]A. A. Albert: “Almost alternative algebras,” Port. Math. 8 (1949), pp. 23–36. MR 0032609 Zbl 0033.15401 article

[139]A. A. Albert: Solid analytic geometry. McGraw-Hill (New York), 1949. Reprinted by Univ. Chicago press in 1966. book

[140]A. A. Albert: “A theory of power-associative commutative algebras,” Trans. Am. Math. Soc. 69 : 3 (November 1950), pp. 503–527. MR 0038959 Zbl 0039.26501 article

[141]A. A. Albert: “A note on the exceptional Jordan algebra,” Proc. Nat. Acad. Sci. U. S. A. 36 : 7 (July 1950), pp. 372–374. MR 0035753 Zbl 0037.15904 article

An associative algebra \( \mathfrak{A} \) over a field \( \mathfrak{F} \) is a vector space over \( \mathfrak{F} \) together with an associative bilinear operation \( xy \). When the characteristic of \( \mathfrak{F} \) is not two we can use the same vector space and define a new algebra \( \mathfrak{A}^{(+)} \) relative to the operation \( \frac{1}{2}(xy + yx) \). This algebra is a Jordan algebra. Any Jordan algebra \( \mathfrak{J} \) is called a special Jordan algebra if \( \mathfrak{J} \) is isomorphic to a Jordan subalgebra of some \( \mathfrak{A}^{(+)} \).

In 1934 it was shown that the Jordan algebra \( \mathfrak{G} \) of all three-rowed Hermitian matrices with elements in the simple eight-dimensional Cayley algebra \( \mathfrak{C} \) is exceptional in the limited sense that \( \mathfrak{G} \) is not isomorphic to a subalgebra of a finite-dimensional \( \mathfrak{A}^{(+)} \). In the present note we shall give a simpler proof of the fact that \( \mathfrak{G} \) is not a special Jordan algebra and shall delete the restriction that \( \mathfrak{A} \) be finite dimensional. We shall assume that \( \mathfrak{G} \) is imbedded in an associative algebra \( \mathfrak{A} \) and shall then obtain a contradiction by showing that the enveloping associative algebra of \( \mathfrak{G} \) contains a subalgebra isomorphic to the non-associative algebra \( \mathfrak{T} \).

[142]A. A. Albert: “New simple power-associative algebras,” Summa Brasil. Math. 2 (1951), pp. 183–194. MR 0048421 Zbl 0045.32101 article

[143]A. A. Albert: “On simple alternative rings,” Can. J. Math. 4 : 2 (1952), pp. 129–135. MR 0048420 Zbl 0046.25403 article

The only known simple alternative rings which are not associative are the Cayley algebras. Every such algebra has a scalar extension which is isomorphic over its center \( \mathsf{F} \) to the algebra \[ \mathsf{C} = e_{11}\mathsf{F} + e_{00}\mathsf{F} + \mathsf{C}_{10} + \mathsf{C}_{01} ,\] where \[ \mathsf{C}_{ij} = e_{ij}\mathsf{F} + f_{ij}\mathsf{F} + g_{ij}\mathsf{F} \qquad (i,j = 0,1;\ i \neq j) .\] The elements \( e_{11} \) and \( e_{00} \) are orthogonal idempotents and \[ e_{ii}x_{ij} = x_{ij}e_{jj} = x_{ij},\quad e_{jj}x_{ij} = x_{ij}e_{ii} = 0,\quad x_{ij}^2 = 0 \] for every \( x_{ij} \) of \( \mathsf{C}_{ij} \). The multiplication table of \( \mathsf{C} \) is then completed by the relations \begin{gather*} f_{10}g_{10} = e_{01},\quad g_{10}e_{10} = f_{01},\quad e_{10}f_{10} = g_{01},\\ g_{01}f_{01} = e_{10},\quad e_{01}g_{01} = f_{10},\quad f_{01}e_{01} = g_{10},\\ e_{ij}e_{ji} = f_{ij}f_{ji} = g_{ij}e_{ji} = e_{ii},\\ e_{ij}f_{ji} = e_{ij}g_{ji} = f_{ij}e_{ji} = f_{ij}g_{ji} = g_{ij}e_{ij} = g_{ij}f_{ij} = 0. \end{gather*}

Every simple alternative ring which contains an idempotent not its unity quantity is either associative or is the Cayley algebra \( \mathsf{C} \).

[144]A. A. Albert: “Power-associative algebras,” pp. 25–32 in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 30 August–6 September, 1950), vol. 2. American Mathematical Society (Providence, RI), 1952. MR 0045103 Zbl 0049.02401 incollection

[145]A. A. Albert: “On nonassociative division algebras,” Trans. Am. Math. Soc. 72 (1952), pp. 296–309. MR 0047027 Zbl 0046.03601 article

[146]A. A. Albert: “On commutative power-associative algebras of degree two,” Trans. Am. Math. Soc. 74 (1953), pp. 323–343. MR 0052409 Zbl 0051.02302 article

The study of the structure of any simple commutative power-associative algebra \( \mathfrak{A} \) over a field \( \mathfrak{F} \) begins with the case where \( \mathfrak{F} \) is algebraically closed. We shall also assume that the characteristic of \( \mathfrak{F} \) is prime to 30. Then \( \mathfrak{A} \) has a unity quantity 1 which is expressible as the sum \[ 1=u_1+\cdots+u_t \] of pairwise orthogonal (absolutely) primitive idempotents \( u_i \). The integer \( t \) is unique and is called the degree of \( \mathfrak{A} \). It is known that if \( t > 2 \), the algebra \( \mathfrak{A} \) is a classical Jordan algebra. The structure of algebras of degree two is not known and will be considered here.

[147]A. A. Albert: “Rational normal matrices satisfying the incidence equation,” Proc. Am. Math. Soc. 4 (1953), pp. 554–559. MR 0056570 Zbl 0050.01004 article

[148]A. A. Albert: “The structure of right alternative algebras,” Ann. Math. (2) 59 : 3 (May 1954), pp. 408–417. MR 0061096 Zbl 0055.26501 article

An algebra \( \mathfrak{A} \) over a field \( \mathfrak{F} \) is said to be right alternative if \( (yx)x = y(xx) \) for every \( x \) and \( y \) of \( \mathfrak{A} \). Some properties of right alternative agebras have been obtained, and it is known that every simple right alternative algebra of characteristic zero is alternative.

In the present paper we shall derive a structure theory for right alternative algebras over any field \( \mathfrak{F} \) of characteristic not two. We shall define a trace function for such algebras and shall show that this trace function has the properties which imply that the ordinary trace criterion for the radical is valid. Our results will then lead to the theorem which states that every semisimple right alternative algebra of characteristic not two is alternative.

[149]A. A. Albert and M. S. Frank: “Simple Lie algebras of characteristic \( p \),” Rend. Sem. Mat. Univ. Politec. Torino 14 (1954/1955), pp. 117–139. MR 0079222 Zbl 0065.26801 article

[150]A. A. Albert: “Leonard Eugene Dickson, 1874–1954,” Bull. Am. Math. Soc. 61 (1955), pp. 331–345. MR 0070564 Zbl 0065.24408 article

[151]A. A. Albert: “On Hermitian operators over the Cayley algebra,” Proc. Nat. Acad. Sci. U. S. A. 41 : 9 (September 1955), pp. 639–640. MR 0071416 Zbl 0065.02002 article

[152]A. A. Albert: “On involutorial algebras,” Proc. Nat. Acad. Sci. U.S.A. 41 : 7 (July 1955), pp. 480–482. MR 0070623 Zbl 0064.27201 article

[153]A. A. Albert: “A property of special Jordan algebras,” Proc. Nat. Acad. Sci. U.S.A. 42 : 9 (September 1956), pp. 624–625. MR 0081271 Zbl 0071.25803 article

[154]A. A. Albert: “A property of ordered rings,” Proc. Am. Math. Soc. 8 (1957), pp. 128–129. MR 0082480 Zbl 0081.26401 article

[155]A. A. Albert: “On certain trinomial equations in finite fields,” Ann. Math. (2) 66 : 1 (July 1957), pp. 170–178. MR 0087691 Zbl 0082.03702 article

Let \( \mathfrak{F} = \mathfrak{F}_q \) be a field of \( q = \pi^m \) elements where the prime \( \pi \) is the characteristic of \( \mathfrak{F} \), and \( \mathfrak{K} \) be a field of degree \( n \) over \( \mathfrak{F} \). If \( y \) is a primitive element of \( \mathfrak{K} \) its period (order as an element of the multiplicative group \( \mathfrak{K}^* \) of all non-zero elements of \( \mathfrak{K} \)) is \( q^n - 1 \), and every element of \( \mathfrak{K}^* \) is a power of \( y \). It follows that if \( s \) is any integer not divisible by \( q^n - 1 \) there exists an integer \( t \) such that \begin{equation*}\tag{1} x^s + y^t = 1. \end{equation*} When \( y \) is not primitive there may be no solution pair \( (s,t) \) of equation (1). Indeed it is easy to show that no solution pair exists in the case where the period of \( y \) is a prime \( p > 3 \) such that \( q \) is primitive modulo \( p \). It is then of some interest to study the question of the existence of solution pairs for some special values of the element \( y \).

In this paper we shall consider the case where \( y = x - 1 \) for \( x \) an element of prime period \( p > 3 \) such that \( q \) is primitive modulo \( p \). We shall restrict our attention to those solution pairs with \( s \) and \( t \) prime to \( p \) and shall determine all such solutions. We shall also separate the solutions into classes relative to conjugacy.

[156]A. A. Albert and B. Muckenhoupt: “On matrices of trace zero,” Michigan Math. J. 4 : 1 (1957), pp. 1–3. MR 0083961 Zbl 0077.24304 article

[157]A. A. Albert: “On partially stable algebras,” Trans. Am. Math. Soc. 84 : 2 (1957), pp. 430–443. MR 0092778 Zbl 0077.25603 article

[158]A. A. Albert: “The norm form of a rational division algebra,” Proc. Nat. Acad. Sci. U.S.A. 43 : 6 (1957), pp. 506–509. MR 0086807 Zbl 0078.02602 article

[159]A. A. Albert and N. Jacobson: “On reduced exceptional simple Jordan algebras,” Ann. Math. (2) 66 : 3 (November 1957), pp. 400–417. MR 0088487 Zbl 0079.04604 article

[160]A. A. Albert: “On the orthogonal equivalence of sets of real symmetric matrices,” J. Math. Mech. 7 (1958), pp. 219–235. MR 0092753 Zbl 0087.01203 article

[161]A. A. Albert: “Finite noncommutative division algebras,” Proc. Am. Math. Soc. 9 (1958), pp. 928–932. MR 0103212 Zbl 0092.03501 article

[162]A. A. Albert and J. Thompson: “Two element generation of the projective unimodular group,” Bull. Am. Math. Soc. 64 (1958), pp. 92–93. MR 0096728 Zbl 0082.02605 article

[163]A. A. Albert: “Addendum to the paper on partially stable algebras,” Trans. Am. Math. Soc. 87 (1958), pp. 57–62. Addendum to paper in Trans. Am. Math. Soc. 84:2 (1957). MR 0092779 Zbl 0083.02603 article

[164]A. A. Albert: Fundamental concepts of higher algebra. University of Chicago Press, 1958. Chapter 5 translated into Russian in Kibern. Sb. 3 (1966). MR 0098735 Zbl 0073.00802 book

[165]A. A. Albert: “A construction of exceptional Jordan division algebras,” Ann. Math. (2) 67 : 1 (January 1958), pp. 1–28. MR 0091946 Zbl 0079.04701 article

We shall construct a class of central simple exceptional Jordan algebras \( \mathfrak{H} \) over any field \( \mathfrak{F} \) of characteristic not two which we shall call cyclic Jordan algebras. Each such algebra \( \mathfrak{H} \) has an attached cyclic associative algebra \( \mathfrak{D} = (\mathfrak{K},S,\tau) \) of degree three over its center \( \mathfrak{F} \), and an attached Cayley algebra \( \mathfrak{C} \) over \( \mathfrak{K} \) having a nonsingular linear transformation \( T \) over \( \mathfrak{F} \) inducing \( S \) in \( \mathfrak{K} \) and such that \[ [(xy)T]\overline{g} = (xT)(yT\bar{g}), \qquad xT^3 = g^{-1}xg \] for \( g = gT \) in \( \mathfrak{C} \). We shall determine \( T \) completely in the case where \( \mathfrak{F}[g] \) is the direct sum of two copies of \( \mathfrak{F} \), and shall show that \( T \) is uniquely determined by a parameter \( \sigma \) in \( \mathfrak{F} \). When \( \mathfrak{D} \) is not a division algebra the cyclic algebra \( \mathfrak{H} \) is reduced. When \( \mathfrak{D} \) is a division algebra we can write \( \mathfrak{H} = \mathfrak{F}(\mathfrak{K},\tau,\sigma) \) and can show that every such \( \mathfrak{H} \) contains a subalgebra of dimension nine over \( \mathfrak{F} \) containing no singular elements. The norm form of \( \mathfrak{D} \) is a cubic form \( \Delta(x) \) in the nine coordinates of the general element \( x \) of \( \mathfrak{D} \), and we shall show that \( \mathfrak{H} \) contains a singular element if and only if \( \sigma = \Delta(x) \) for some \( x \) of \( \mathfrak{D} \). Also \( \mathfrak{H} \) is reduced if and only if \( \sigma = \Delta(x) \). It follows that every simple exceptional Jordan algebra over an algebraic number field is reduced. However, if \( \mathfrak{F}_0 \) is any field of characteristic not two such that there exists a cyclic associative division algebra \( \mathfrak{D}_0 = (\mathfrak{K}_0, S, \tau) \) over \( \mathfrak{F}_0 \), and \( \sigma \) is an indeterminate over \( \mathfrak{F}_0 \), the cyclic Jordan algebra \( \mathfrak{H} = \mathfrak{H}(\mathfrak{K}_{0\mathfrak{F}}, \tau, \sigma) \) over \( \mathfrak{F} = \mathfrak{F}_0(\sigma) \) is not reduced and contains no divisors of zero.

[166]A. A. Albert: “On the collineation groups associated with twisted fields,” pp. 485–497 in Calcutta Mathematical Society Golden Jubilee Commemoration (1958/59), part II. Calcutta Mathematical Society, 1958/1959. MR 0159854 Zbl 0163.28003 incollection

[167]A. A. Albert and L. J. Paige: “On a homomorphism property of certain Jordan algebras,” Trans. Am. Math. Soc. 93 (1959), pp. 20–29. MR 0108524 Zbl 0089.02001 article

[168]A. A. Albert and J. Thompson: “Two-element generation of the projective unimodular group,” Ill. J. Math. 3 : 3 (1959), pp. 421–439. MR 0106951 Zbl 0098.02302 article

Let \( \mathfrak{F} = \mathfrak{F}_q \) be the field of \( q = p^m \) elements, \( \mathfrak{M} = \mathfrak{M}(n,q) \) the multiplicative group of all \( n \)-rowed square matrices with elements in and determinant 1, and \( \mathfrak{N} = \mathfrak{N}(n,q) \) the subgroup of \( \mathfrak{M} \) consisting of its scalar matrices \( \rho I \) with \( \rho^n = 1 \). We assume, of course, that \( n > 1 \). Then \( \mathfrak{N} \) is a normal subgroup of \( \mathfrak{N} \), and the quotient group \[ \mathfrak{G}=\mathfrak{G}(n,q)=\mathfrak{M}/\mathfrak{N} \] is a well-known simple group called the projective unimodular group.

In [1930] H. R. Brahana gave a list of simple groups of orders less than 1,000,000. An examination of his list reveals the fact that every group there is generated by two elements, one of which has period (group order) two. The purpose of this paper is to prove the corresponding result for a general class of simple groups. We shall derive the following property.

The projective unimodular group is generated by two elements \( A\mathfrak{N} \) and \( B\mathfrak{N} \), where the coset \( A\mathfrak{N} \) has period two.

[169]A. A. Albert: “A solvable exceptional Jordan algebra,” J. Math. Mech. 8 (1959), pp. 331–337. MR 0101875 Zbl 0204.04101 article

[170]A. A. Albert: “On the collineation groups of certain non-desarguesian planes,” Port. Math. 18 (1959), pp. 207–224. MR 0130613 Zbl 0099.15203 article

[171]Finite groups (New York, 23–24 April, 1959). Edited by A. A. Albert and I. Kaplansky. Proceedings of Symposia in Pure Mathematics 1. American Mathematical Society (Providence, RI), 1959. Zbl 0097.25602 book

[172]A. A. Albert: “Finite division algebras and finite planes,” pp. 53–70 in Combinatorial analysis (Columbia University, New York, April 24–26, 1958). Edited by R. E. Bellman and M. Hall, Jr. Proceedings of Symposia in Applied Mathematics 10. American Mathematical Society (Providence, RI), 1960. MR 0116036 Zbl 0096.15003 incollection

Richard Ernest Bellman	Related
Marshall Hall, Jr.	Related

[173]A. A. Albert: “Generalized twisted fields,” Pac. J. Math. 11 : 1 (1961), pp. 1–8. MR 0122850 Zbl 0154.27203 article

[174]A. A. Albert: Structure of algebras, 7th edition. AMS Colloquium Publications 24. American Mathematical Society (Providence, RI), 1961. Revised printing. MR 0123587 book

[175]A. A. Albert: “Isotopy for generalized twisted fields,” Anais Acad. Brasil. Ci. 33 (1961), pp. 265–275. MR 0139639 Zbl 0154.27301 article

[176]A. A. Albert: “Finite planes for the high school,” Math. Teach. 55 : 3 (March 1962), pp. 165–169. An address given in connection with the Thirty-ninth Annual Meeting of the National Council of Teachers of Mathematics, Chicago, 7 April, 1961. article

[177]A. A. Albert: “On the nuclei of a simple Jordan algebra,” Proc. Nat. Acad. Sci. U.S.A. 50 : 3 (1963), pp. 446–447. MR 0153718 Zbl 0115.02702 article

[178]A. A. Albert: “On involutorial associative division algebras,” Scripta Math. 26 (1963), pp. 309–316. MR 0179202 Zbl 0147.28702 article

[179]Studies in modern algebra. Edited by A. A. Albert. MAA Studies in Mathematics 2. Prentice-Hall (Englewood Cliffs, NJ), 1963. MR 0146228 Zbl 0192.00102 book

[180]A. A. Albert: “On associative division algebras of prime degree,” Proc. Am. Math. Soc. 16 (1965), pp. 799–802. MR 0179203 Zbl 0135.07402 article

In 1938 Richard Brauer showed that, if \( \mathfrak{D} \) is an associative division algebra of degree five over its center \( \mathfrak{F} \), there is a field \( \mathfrak{K} \) of degree at most twelve over \( \mathfrak{F} \) such that the scalar extension \( \mathfrak{D}\times\mathfrak{K} \) is a cyclic algebra over \( \mathfrak{K} \). Indeed \[ \mathfrak{K}=\mathfrak{F}(\alpha_1,\alpha_2,\alpha_3) ,\] where \( \alpha_1 \) is a root of a quadratic equation over \( \mathfrak{F} \), \( \alpha_2 \) is a root of a quadratic equation over \( \mathfrak{F}(\alpha_1) \), and \( \alpha_3 \) is a root of a cubic equation over \( \mathfrak{F}(\alpha_1,\alpha_2) \). Since that time there has been no progress in the study of the structure of associative central division algebras of prime degree.

In view of Brauer’s result it seems reasonable, as a first step in the study of central division algebras \( \mathfrak{D} \) of prime degree \( p \) over \( \mathfrak{F} \), to consider the case where there is a quadratic field \( \mathfrak{K} \) over \( \mathfrak{F} \) such that \( \mathfrak{D}_{\mathfrak{K}} \) is cyclic. Then \[ \mathfrak{D}_{\mathfrak{K}}=\mathfrak{D}\times\mathfrak{K} \] has a subfield \( \mathfrak{Z} \) which is cyclic of degree \( p \) over \( \mathfrak{F} \). The simplest subcase is that where \( \mathfrak{Z} \) is actually normal, but, of course, not cyclic over \( \mathfrak{F} \). We shall treat this case under the assumption that \( \mathfrak{F} \) has characteristic \( p \), and shall prove that then \( \mathfrak{D} \) is a cyclic algebra over \( \mathfrak{F} \).

[181]A. A. Albert: “A normal form for Riemann matrices,” Canad. J. Math. 17 (1965), pp. 1025–1029. MR 0182628 Zbl 0196.30104 article

[182]A. A. Albert: “On exceptional Jordan division algebras,” Pac. J. Math. 15 : 2 (1965), pp. 377–404. MR 0182647 Zbl 0135.07202 article

In 1957 the author gave a construction of a class, of central simple exceptional Jordan algebras \( \mathfrak{H} \), over any field \( \mathfrak{F} \) of characteristic not two, called cyclic Jordan algebras. The principal ingredients of this construction were the following:

A cyclic cubic field \( \mathfrak{K} \) with generating automorphism \( S \) over \( \mathfrak{F} \).
A Cayley algebra \( \mathfrak{C} \), with \( \mathfrak{K} \) as center, so that \( \mathfrak{C} \) has dimension eight over \( \mathfrak{K} \), and dimension 24 over \( \mathfrak{F} \).
A nonsingular linear transformation \( T \) over \( \mathfrak{F} \) of \( \mathfrak{C} \), which induces \( S \) in \( \mathfrak{K} \), and commutes with the conjugate operation of \( \mathfrak{C} \).
An element \( g \) in \( \mathfrak{C} \), and a nonzero element \( \gamma \) of \( \mathfrak{K} \), such that \[ g=gT \quad\text{and}\quad g\bar{g} = [\gamma(\gamma S)(\gamma S^2)]^{-1} .\] Thus \( g \) is nonsingular. Also the polynomial algebra \( \mathfrak{G}=\mathfrak{F}[g] \) is either a quadratic field over \( \mathfrak{F} \) or is the direct sum, \[ \mathfrak{G}=e_1\mathfrak{F}\oplus e_2\mathfrak{F} ,\] of two copies \( e_i\mathfrak{F} \) of \( \mathfrak{F} \).
The properties \[ [g(xy)T]=[g(xT)](yT) \quad\text{and}\quad xT^3 = g^{-1}xg ,\] for every \( x \) and \( y \) of \( \mathfrak{C} \).

In the present paper we shall give a general solution of the equations of (v), and shall determine \( T \) in terms of two parameters in \( \mathfrak{L}=\mathfrak{K}[g] \) satisfying some conditions of an arithmetic type. We shall also provide a special set of values of all of the parameters of our construction, and shall so provide a proof of the existence of cyclic Jordan division algebras with attached Cayley algebra \( \mathfrak{C} \) a division algebra.

The existence of a transformation \( T \) with the two properties of (v) for some element \( g = gT \), in the Cayley algebra \( \mathfrak{C} \) which satisfies (iv), was demonstrated by the author in the paper [Albert 1958] only in the case where \( \mathfrak{G} \) is not a field, and consequently \( \mathfrak{C} \) is a split algebra. In that case it was proved that cyclic Jordan division algebras do exist, for certain kinds of fields \( \mathfrak{F} \). Thus the case where \( \mathfrak{G} \) is a field, and \( \mathfrak{C} \) may possibly be a division algebra, remained.

[183]A. A. Albert: “On some properties of biabelian fields,” Anais Acad. Brasil. Ci. 38 (1966), pp. 217–221. MR 0207683 Zbl 0154.03903 article

[184]A. A. Albert: “The finite planes of Ostrom,” Bol. Soc. Mat. Mex., II. Ser. 11 (1966), pp. 1–13. MR 0231271 Zbl 0157.27003 article

[185]A. A. Albert: “Finite fields,” Kibern. Sb., Nov. Ser. 3 (1966), pp. 7–49. Russian translation of chapter 5 of Fundamental concepts of higher algebra (1956). Zbl 0147.01802 article

[186]A. A. Albert: “On certain polynomial systems,” Scripta Math. 28 (1967), pp. 15–19. MR 0211990 Zbl 0152.02302 article

[187]A. A. Albert: “New results on associative division algebras,” J. Algebra 5 : 1 (January 1967), pp. 110–132. MR 0202757 Zbl 0144.02503 article

It is well known that every central division algebra of degree two or three is a cyclic algebra, and that every algebra of degree four is a crossed product. It is also known that there exist noncyclic algebras of degree four. Very little is known about algebras of degree \( n > 4 \).

The principal question in the theory of associative division algebras is then the question of the existence of algebras which are not crossed products. It is thus natural to discuss algebras of prime degree \( p \), a case where every crossed product is cyclic.

We shall address ourselves here to a study of the nature of division algebras \( \mathfrak{D} \) of prime degree \( p \) over a field \( \mathfrak{F} \) of characteristic \( p \), and shall limit our discussion to the case where there exists a quadratic extension \( \mathfrak{K} \) of \( \mathfrak{F} \) such that the extension \( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \) is a cyclic algebra.

[188]A. A. Albert: “On associative division algebras,” Bull. Am. Math. Soc. 74 (1968), pp. 438–454. Retiring Presidential Address delivered at the 74th annual meeting of the AMS on January 23, 1967, in San Francisco. MR 0222114 Zbl 0157.08001 article

We shall study the structure of a central division algebra \( \mathfrak{D} \), of odd prime degree \( p \) over any field \( \mathfrak{F} \) of characteristic \( p \), which has the property that there exists a quadratic extension field \( \mathfrak{K} \) of \( \mathfrak{F} \) such that the algebra \( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \) is cyclic over \( \mathfrak{K} \). We shall obtain a simplified version of the condition that a cyclic algebra \( \mathfrak{D}_0 \), of degree \( p \) over \( \mathfrak{K} \), shall possess the factorization property \( \mathfrak{D}_0 = \mathfrak{D}\times\mathfrak{K} \). We shall also derive a new sufficient condition that such a \( \mathfrak{D} \) shall be cyclic over \( \mathfrak{F} \), and shall present a large class of our algebras \( \mathfrak{D}_0 \) which satisfy this condition.

[189]A. A. Albert and R. Sandler: An introduction to finite projective planes. Holt, Rinehart and Winston (New York), 1968. MR 0227851 Zbl 0194.21502 book

[190]A. A. Albert: “A note on certain cyclic algebras,” J. Algebra 14 : 1 (September 1970), pp. 70–72. MR 0251066 Zbl 0186.06901 article

[191]A. A. Albert: “Tensor products of quaternion algebras,” Proc. Am. Math. Soc. 35 : 1 (September 1972), pp. 65–66. MR 0297803 Zbl 0263.16012 article

[192]Collection of articles dedicated to the memory of Abraham Adrian Albert, published as Scripta Math. 29 : 3–4. Yeshiva University (New York), 1973. MR 0396137 book

[193]I. N. Herstein: “On a theorem of A. A. Albert,” pp. 391–394 in Collection of articles dedicated to the memory of Abraham Adrian Albert, published as Scripta Math. 29 : 3–4 (1973). MR 0435137 Zbl 0266.16014 incollection

[194]I. N. Herstein: “A. Adrian Albert,” pp. iv–v in Collection of articles dedicated to the memory of Abraham Adrian Albert, published as Scripta Math. 29 : 3–4 (1973). MR 0403899 incollection

[195]D. Zelinsky: “A. A. Albert,” Am. Math. Monthly 80 (1973), pp. 661–665. MR 0313017 Zbl 0264.01019 article

[196]N. Jacobson: “Abraham Adrian Albert (1905–1972),” Bull. Am. Math. Soc. 80 (1974), pp. 1075–1100. MR 0342341 Zbl 0292.01025 article

[197]L. E. Dickson: The collected mathematical papers of Leonard Eugene Dickson, vol. 3. Edited by A. A. Albert. Chelsea Publishing Co. (New York), 1975. MR 0441667 book

[198]L. E. Dickson: The collected mathematical papers of Leonard Eugene Dickson, vol. 5. Edited by A. A. Albert. Chelsea Publishing Co. (New York), 1975. MR 0441669 book

[199]L. E. Dickson: The collected mathematical papers of Leonard Eugene Dickson, vol. 4. Edited by A. A. Albert. Chelsea Publishing Co. (New York), 1975. MR 0441668 book

[200]L. E. Dickson: The collected mathematical papers of Leonard Eugene Dickson, vol. 1. Edited by A. A. Albert. Chelsea Publishing Co. (New York), 1975. MR 0441665 book

[201]L. E. Dickson: The collected mathematical papers of Leonard Eugene Dickson, vol. 2. Edited by A. A. Albert. Chelsea Publishing Co. (New York), 1975. MR 0441666 book

[202]L. E. Dickson: The collected mathematical papers of Leonard Eugene Dickson, vol. 6. Edited by A. A. Albert. Chelsea Publishing Co. (New York), 1983. MR 749229 book

[203]I. Kaplansky: “Abraham Adrian Albert: November 9, 1905–June 6, 1972,” pp. 244–264 in A century of mathematics in America, part I. Edited by P. Duren, R. A. Askey, and U. C. Merzbach. History of Mathematics 1. American Mathematical Society (Providence, RI), 1988. MR 1003173 Zbl 0667.01014 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Irving Kaplansky	Related	Volume
Richard Allen Askey	Related	Volume

[204]A. A. Albert: Collected mathematical papers, part 2: Nonassociative algebras and miscellany. Edited by R. E. Block, N. Jacobson, J. M. Osborn, D. J. Saltman, and D. Zelinsky. Collected Works 3.2. American Mathematical Society (Providence, RI), 1993. With a preface by the editors and Irving Kaplansky. With biographies of Albert by Zelinsky, Jacobson, Kaplansky and Israel N. Herstein. MR 1213452 Zbl 0790.01040 book

David J. Saltman	Related
Daniel Zelinsky	Related
Richard Earl Block	Related
Irving Kaplansky	Related	Volume
Israel N. Herstein	Related
Nathan Jacobson	Related
J. Marshall Osborn	Related

[205]A. A. Albert: Collected mathematical papers, part 1: Associative algebras and Riemann matrices. Edited by R. E. Block, N. Jacobson, J. M. Osborn, D. J. Saltman, and D. Zelinsky. Collected Works 3.1. American Mathematical Society (Providence, RI), 1993. With a preface by the editors and Irving Kaplansky. With biographies of Albert by Zelinsky, Jacobson, Kaplansky and Israel N. Herstein. MR 1213451 Zbl 0790.01039 book

David J. Saltman	Related
Daniel Zelinsky	Related
Richard Earl Block	Related
Irving Kaplansky	Related	Volume
Israel N. Herstein	Related
Nathan Jacobson	Related
J. Marshall Osborn	Related

[206]N. E. Albert: A\( ^3 \) & his algebra: How a boy from Chicago’s west side became a force in American mathematics. iUniverse (Lincoln, NE), 2005. With a preface by Erwin Kleinfeld. MR 2147379 Zbl 1069.01006 book

Nancy E. Albert	Related
Erwin Kleinfeld	Related

[207]D. D. Fenster and J. Schwermer: “A delicate collaboration: Adrian Albert and Helmut Hasse and the principal theorem in division algebras in the early 1930s,” Arch. Hist. Exact Sci. 59 : 4 (2005), pp. 349–379. MR 2188938 Zbl 1066.01026 article

Traditionally, the words “collaboration,” and “principal theorem in division algebras in the 1930s” are associated with the celebrated German trio of mathematicians, Richard Brauer, Helmut Hasse and Emmy Noether. In fact, Brauer, Hasse, and Noether formed one of the collaborative efforts that led to the proof of the principal theorem in linear algebras in the 1930s, that is, the structural description of normal division algebras over an algebraic number field. This paper, however, highlights the other joint work linked with the proof of this theorem, namely that of A. Adrian Albert and Hasse. The paper’s title alludes to a tension in this collaboration. It also refers to the delicate rapport Hasse and Albert cultivated in their correspondence. The seemingly cordial friendship they projected on paper helped keep potentially difficult issues in balance and, consequently, preserved their mutual exchanges. Our specific avenue of investigation emphasizes the correspondence from Albert to Hasse in 1931 and early 1932. At that time, Brauer, Hasse and Noether worked to prove the principal theorem in Germany. Ultimately, in late 1931, this triumvirate established the principal theorem that every normal division algebra over an algebraic number field of finite degree is cyclic [Brauer et al. 1932]. Two months later, Albert and Hasse published a joint work with an alternative proof, entitled “A Determination of all Normal Division Algebras Over an Algebraic Number Field” which also included something of a brief history of the theorem [Albert and Hasse 1932].

In the current paper, we use new archival materials to begin to unravel the behind-the-scenes history of this truly international development in the history of mathematics. In the first two sections of this paper we introduce the protagonists of our research: Albert and Hasse. We trace the individual mathematical paths Albert and Hasse each traveled to arrive at the central question in division algebras. In particular, we see the strong influence of Leonard Eugene Dickson and Joseph Henry Maclagan Wedderburn on the mathematical work of Albert and the prominent role of Kurt Hensel’s \( p \)-adic number theory in Hasse’s mathematical researches. A more formal discussion of the mathematical concepts related to the theory of normal simple algebras in the late 1920s and early 1930s forms the focus of the following section. With this framework in place, we turn our attention to the actual correspondence from Albert to Hasse. These letters contribute to a clearer understanding of the development of a proof of the “principal theorem” in the theory of linear algebras in those waning months of 1931 in Germany and Chicago. Finally, we offer some reflective, concluding remarks on this history.

Joachim Schwermer	Related
Della Dumbaugh Fenster	Related
Helmut Hasse	Related

[208]D. D. Fenster: “Research in algebra at the University of Chicago: Leonard Eugene Dickson and A. Adrian Albert,” pp. 179–197 in Episodes in the history of modern algebra (1800–1950). Edited by J. Gray and K. H. Parshall. History of Mathematics 32. American Mathematical Society (Providence, RI), 2007. MR 2353496 Zbl 1128.01023 incollection

Della Dumbaugh Fenster	Related
L. E. Dickson	Related	Volume
Karen V. Hunger Parshall	Related
Jeremy J. Gray	Related

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