Transactions of the American Mathematical Society, Vol. 14, No. 4 (Oct. 1913), 489-500

Introduction

Consider a system of forms f_{1}(x_{1},...,x_{m}), ..., f_{k}(x_{1},...,x_{m}), whose coefficients a_{1},...,a_{r} (arranged in a definite order) are independent variables. Let b_{1},...,b_{r} be the coefficients (taken in the same order) of the forms derived from f_{1},...,f_{k} by applying a given linear transformation T with integral coefficients. Let F(a_{1},...,a_{r}) be a polynomial with integral coefficients and

D(a_{1},...,a_{r}) = F(b_{1},...,b_{r}) – F(a_{1},...,a_{r})

be the polynomial in a_{1},...,a_{r} with integral coefficients which is obtained from F(b) – F(a) upon replacing b_{1},...,b_{r} by their expressions in terms of a_{1},...,a_{r} and the coefficients of T. in case two or more of the terms in the initial expression for D had the same literal part a_{1}^{e1}···a_{r}^{er}, such terms are assumed to have been combined additively into a single term. Let p be a prime. According to a definition by Hurwitz (stated for a single form in two variables), F(a_{1},...,a_{r}) is an invariant of f_{1},...,f_{k} modulo p with respect to the transformation T if D(a_{1},...,a_{r}) = 0 (mod p), identically in a_{1},...,a_{r}, viz., if the coefficient of each term of D is divisible by p. He gave an interesting example of such an invariant with respect to all of the transformations T with integral coefficients whose determinant is not divisible by p, but did not construct a theory of invariants modulo p.

As a generalization we may employ transformations T with coefficients in the Galois field DF[p^{n}] composed of the p^{n} elements

c_{0} + c_{1}j + c_{2}j^{2} + c_{n-1}j^{n-1} (c_{0},...,c_{n-1} = 0,1,...,p-1),

where j is a root of a congruence of degree n irreducible modulo p. A polynomial F(a_{1},...,a_{r}) with integral coefficients is a *formal* invariant of f_{1},...,f_{k} in the field, under transformation T if D(a_{1},...,a_{r}) is identically zero in the field a_{1},...,a_{r}, i.e., if the coefficient of each term of D, when reduced to the form c_{0} + c_{1}j + c_{2}j^{2} + c_{n-1}j^{n-1}by means of the congruence satisfied by j, has each c_{i} a multiple of p. If F_{0}, F_{1},... are such formal invariants (with integral coefficients), then F_{0} + F_{1}j +... is a formal invariant. In this manner, or by direct extension of the previous definition, we have the concept of a formally invariant polynomial with coefficients in the GF[p^{n}].

We pass to the entirely different concept of *modular* invariants, introduced by Dickson. The coefficients a_{1},...,a_{r} of the forms are now undetermined elements of the GF[p^{n}]. A polynomial F(a_{1},...,a_{r}) with coefficients in that field is called a modular invariant of the system of forms under any given group G of linear transformations with coefficients in the field if, for each transformation of G, D(a_{1},...,a_{r}) is zero in the field. To apply this test we may first express D as a polynomial δ(a_{1},...,a_{r}) in which the exponents are all less than p^{n}, and then require that δ shall be identically zero in the field as to a_{1},...,a_{r}. We thus see clearly just how the difference in the definitions of formal and modular invariants affects the actual computations. Dickson has given a very simple and elegant theory of modular invariants. No theory has been developed for formal invariants. However, there exists between the two subjects an interesting and important relation, which I shall develop in what follows. I take this opportunity to express my gratitude to Professor Dickson for his interest and many helpful suggestions, in particular for the present formulation of this introductory section.

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