Consider a system of forms f1(x1,...,xm), ..., fk(x1,...,xm), whose coefficients a1,...,ar (arranged in a definite order) are independent variables. Let b1,...,br be the coefficients (taken in the same order) of the forms derived from f1,...,fk by applying a given linear transformation T with integral coefficients. Let F(a1,...,ar) be a polynomial with integral coefficients and
D(a1,...,ar) = F(b1,...,br) — F(a1,...,ar)
be the polynomial in a1,...,ar with integral coefficients which is obtained from F(b) — F(a) upon replacing b1,...,br by their expressions in terms of a1,...,ar and the coefficients of T. in case two or more of the terms in the initial expression for D had the same literal part a1e1···arer, 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(a1,...,ar) is an invariant of f1,...,fk modulo p with respect to the transformation T if D(a1,...,ar) = 0 (mod p), identically in a1,...,ar, 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[pn] composed of the pn elements
c0 + c1j + c2j2 + cn-1jn-1 (c0,...,cn-1 = 0,1,...,p-1),
where j is a root of a congruence of degree n irreducible modulo p. A polynomial F(a1,...,ar) with integral coefficients is a formal invariant of f1,...,fk in the field, under transformation T if D(a1,...,ar) is identically zero in the field a1,...,ar, i.e., if the coefficient of each term of D, when reduced to the form c0 + c1j + c2j2 + cn-1jn-1 by means of the congruence satisfied by j, has each ci a multiple of p. If F0, F1,... are such formal invariants (with integral coefficients), then F0 + F1j +... 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[pn].
We pass to the entirely different concept of modular invariants, introduced by Dickson. The coefficients a1,...,ar of the forms are now undetermined elements of the GF[pn]. A polynomial F(a1,...,ar) 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(a1,...,ar) is zero in the field. To apply this test we may first express D as a polynomial δ(a1,...,ar) in which the exponents are all less than pn, and then require that δ shall be identically zero in the field as to a1,...,ar. 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.