Celebratio Mathematica

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

G. A. Bliss: “Eliakim Hastings Moore,” Bull. Amer. Math. Soc. 39 : 11 (1933), pp. 831–838. MR 1562740 JFM 59.0038.02

G. A. Bliss	Related	Volume
E. H. Moore	Related	Volume

G. A. Bliss and L. E. Dickson: “Eliakim Hastings Moore: 1862–1932,” Biographical Memoirs of the National Academy of Sciences 17 (1937), pp. 83–102. article

G. A. Bliss	Related	Volume
E. H. Moore	Related	Volume

L. E. Dickson: Linear groups with an exposition of the Galois field theory, reprint edition. Dover Publications (New York), 1958. With an introduction by Wilhelm Magnus. Republication of 1901 original. MR 0104735 Zbl 0082.24901 book

L. E. Dickson: Linear algebras, reprint edition. Hafner (New York), 1960. Republication of 1914 original. MR 0118745 book

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Leonard Eugene Dickson

Role modeling in mathematics:
The case of Leonard Eugene Dickson (1874–1954)

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Role modeling in mathematics: The case of Leonard Eugene Dickson (1874–1954)

Filter the Bibliography List

Role modeling in mathematics:
The case of Leonard Eugene Dickson (1874–1954)