L. E. Dickson

A fundamental system of invariants of the general modular linear group with a solution of the form problem

1. WYe shall determine m functions which form a fundamental system of invariants for the group Gm of all linear homogeneous transformations on ns variables with coefficients in the Galois field of order pn. In the so-called form problem for the group G^, we seek all sets of values of the m variables for which the m fundamental absolute invariants take assigned values. It is shown in § 8 that all sets of solutions are linear combinations of the roots of an equation involving only the powers pn pn(n-l) ... pn 1 of a single variable. This fundamental equation has properties analogous to those of a linear differential equation of the m-th order. In §§10-16 we determine the degrees of the irreducible factors of the fundamental equation and, in particular, the smallest field in which it is completely solvable. VVe obtain a wide generalization of the theory of the equation {pnm _ { = 0 which forms the basis of the theory of finite fields. The function defined by the left member of the fundamental equation includes the type of sllbstitlltion quantics in one variable the theory of which is equivalent to, but preceded historically, the theory of linear modular substitutions on m variables. V\re here find that the latter theory necessitates a return to the earlier quantics in one variable. Finally, in §§17-22, we consider the interpletation of certain invariants. It follows from the theorem collcerning the product of two determinants that a trallsformation T of Gm replaces the fulletion

Linear algebras in which division is always uniquely possible

does not vanish. Hence the condition that right hand [left baiid] division shall always be uniquely possible is that A,. [Al] shall vanish only when every a. vanishes. Now either of these conditions is satisfied when the other is, since either is equivalent to the condition that a product shall vanish only when one factor vanishes. We consider algebras in which these conditions are satisfied and in which there is a modulus, i. e., an element 1 such that 1A = Al = A for every element A. We shall henceforth set el = 1. For m = 2, the algebra is the field F(e2). Indeed, e2 _ e2y222 0 is irreducible in F since Ar al + al a2 y222 a a2 y221. In ? 2 I consider the general transformation of algebras with three units, exhibiting families of algebras invariant under every linear transformation and determining the algebras which admit more than one transforination into itself (and hence exactly three transformations). Froin each standpoint I am led to the same remarkable set of families of algebras, each set characterized by a parameter ,u. For , = 1, the family consists of all fields of rank three with respect to F. For u = 0, the commutative algebras have the property that division is always possible.

On commutative linear algebras in which division is always uniquely possible

1. We consider commutative linear algebras in 2n units, with coordinates in a general field F, such that n of the units define a sub-algebra forming a field F( J). The elements of the algebra may be exhibited compactly in the form A + BJ, where A and B range over F(J). As multiplication is not associative in general, A and B do not play the role of coordinates, so that the algebra is not binary in the usual significance of the term.t Nevertheless, by the use of this binary niotation, we may exhibit in a very luminous forml the multiplicatioln-tables of certain algebras in four and six unlits, given in an earlier paper. i Proof of the existence of the algebras and of the uniqtueness of division niow presents no difficulty. The form of the corresponding algebra in 2n units becomes obvious. After thus perfecting and extending known results, we attack the problem of the deternmination of all algebras with the prescribed properties. An extensive new class of algebras is obtained.

Integers represented by positive ternary quadratic forms

Without giving any details, he stated that like considerations applied to the representation of multiples of 3 by B. But the latter problem is much more difficult and no treatment has since been published ; it is solved below by two methods. Ramanujanf readily found all sets of positive integers a, by Cy d such that every positive integer can be expressed in the form ax+by-\-cz+du. He made use of the forms of numbers representable by AyB,C= x + y+2z, D= x+2y+2z} E = x + 2y + 3z9 F = x + 2y + 4z, G = x + 2y + 5z.

Representations of the general symmetric group as linear groups in finite and infinite fields

1. In a series of articles in the Berliner Berichte, beginning in 1896, Frobenius has developed an elaborate theory of group-characters and applied it to the representation of a given finite group Gasa non-modular linear group. Later, Burnside f approached the subject from the standpoint of continuous groups. The writer has shown J that the method employed by Burnside may be replaced by one involving only purely rational processes and hence leading to results valid for a general field. The last treatment, however, expressly excludes the case in which the field has a modulus which divides the order of G. The exclusion of this case is not merely a matter of convenience, nor merely a limitation due to the particular method of treatment ; indeed, § the properties of the group-determinant differ essentially from those holding when the modulus does not divide the order of G.\\ Thus when G is of order q !, the general theory gives no information as to the representations in a field having a modulus = q, whereas the case of a small modulus is the most important one for the applications. The present paper investigates the linear homogeneous groups on m variables, with coefficients in a field F, which are simply isomorphic with the symmetric group on q letters. The treatment is elementary and entirely independent of the papers cited above ; in particular, the investigation is made for all moduli without exception. The principal result is the determination of the minimum value of the number of variables. It is shown that m = q — lormëy2, according as F has not or has a modulus p which divides q (§§ 8-21). There

Linear associative algebras and abelian equations

1. In the recent development of the theory of linear associative algebras the coordinates of whose numbers range over the field of ordinary complex numbers, an important role is played by the so-called simple algebras, the only ones being the matric algebras with r2 units. For the more general case of algebras the coordinates of whose numbers range over any field (Korper) K of finite or infinite order, a corresponding importance is to be attached to simple algebras; an algebra is simple t if and only if it is the direct product of a matric algebra and a primitive algebra (one in which every element has an inverse). The problem of primitive algebras is the chief outstanding problem in the theory of algebras over a field K. Aside from real quaternions the only known primitive linear associative algebras are fields. We proceed to exhibit a primitive algebra L with r2 units over K. Let K (X)-O be a uniserial abelian equation of degree r with respect to the field K, i. e., let the equation have as its coefficients numbers in K, be irreducible in K, and have as its roots

Definitions of a linear associative algebra by independent postulates

The term linear associative algebra, introduced by BENJAMIN PEIRCE, has the same significance as the term system of (higher) complex numbers. t In the usual theory of complex numbers, the co6rdinates 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 co6rdinates belong to an arbitraryfield 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 n3 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.

All integers except 23 and 239 are sums of eight cubes

The algebraic part of Wieferichs proof holds for all integers exceeding 2J billion. The fact that all smaller integers are sums of nine cubes was proved by use of Table I. To prove my theorem, I shall need also the new Tables II and I I I . Table I gives, for each positive integer N^ 40,000, the least number m such that N is a sum of m cubes. I t was computed by R. D. von SterneckJ by adding all cubes to

Singular case of pairs of bilinear, quadratic, or Hermitian forms

The main object of this paper is a treatment of the equivalence of pairs of bilinear forms in the singular case by a purely rational method. The problem was first discussed by Kronecker,t who employed the irrational canonical form due to Weierstrass for the auxiliary non-singular case, instead of the rational canonical form (13) employed here. It is then a simple matter to deduce in Parts II and III the criteria for the equivalence of pairs of symmetric or Hermitian bilinear forms, or quadratic or Hermitian forms, in the singular case.

Canonical forms of quaternary abelian substitutions in an arbitrary Galois field

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