On a theorem of M. Deuring and I. R. Šafarevič

Abstract

Let F/K be a field of algebraic functions of one variable, K algebraically closed of characteristic p≠0. Let E/K be a cyclic extension of F/K of degree ℓ, a prime not necessarily different from p. Let ξ, ρ denote the ℓ-ranks of the null class groups of E and F respectively. If E/F is ramified, Deuring proved that ξ=ℓρ+(t−δ)(ℓ−1) where t is the number of ramified primes and δ is 1 or 2 according as ℓ equals p or not. Šafarevič proved the same relation in the unramified case for ℓ=p where ξ, ρ denote the ranks of the Hasse-Witt matrices. Subrao gave a unified proof for ℓ=p. Rosen and Sullivan have proved the theorem in special cases. In this paper, using galois cohomology, Deuring's method of proof is modified so as to give a proof of the theorem in all cases.