Galois cohomology and class number in constant extension of algebraic function fields

Abstract

Let F be a field of algebraic functions of one variable having the finite field K as exact field of constants. The class number of F is defined as the order of the finite group, Co(F), of divisor classes of degree zero. Let L be the unique cyclic extension of K of degree n, E = F* L the corresponding constant extension with galois group G. Since K is perfect, the canonical homomorphism of the group of divisor classes of F in the group of divisor classes of E is an injection [2, p. 477]. If hE, hF denote the class numbers of E and F, respectively, we have hE=hF. k, for some integer k. The purpose of this note is to prove the following two theorems: