IRREDUCIBLE CONGRUENCES OVER GF(p)

with coefficients belonging to the same field. Two irreducible m-ic congruences are said to belong to the same conjugate set if one of them can be transformed into the other by a transformation of G. The number of distinct irreducible congruences in a conjugate set will be referred to as the order of the conjugate set. Since the order of the group G is p(pi — l), it follows that the order of any conjugate set will be at most p(p2 — l). A classification of the irreducible binary modular forms under the group of all binary linear homogeneous transformations of determinant unity in the field GF(pn) has been done by Dickson[4]. Since an irreducible binary modular form over GF(p) of degree m in x and y defines an irreducible m-ic congruence C(z) over GF(p) with roots \^ = (x/yy\ 7=0, 1, 2, • ■ • , m — 1), in the Galois field GF(pm), it follows that Dickson's results provide a classification of the irreducible m-ic congruences over GF(p) under the subgroup G' of transformations of G with determinant a square in GF(p). Clearly, G' is a proper subgroup of G if p>2, and a conjugate set C under the group G will consist of, at most, two conjugate sets Ci, C{ under the smaller group G', i.e., C = C{\JC2. It is shown in §2 that if aiEGF(pm) characterizes the set C{ under G', then — <ri will characterize the set C2 under G' and a\ will characterize the conjugate set C under the group G. In studying the irreducible binary modular forms over GF(pn), Dickson lists two relative invariants, namely,