Galois groups of global fields of finite characteristic

Abstract

Let G be a finite group having a normal p-Sylow subgroup P such that G P is the direct product of two cyclic groups of orders e and f, both prime to p. Let k be a global field of characteristic p, containing the eth roots of unity, and S a finite set of primes of k. Then there exists a Galois extension L of k with Galois group G( L k ) isomorphic to G, such that the local Galois group G( L v k v ) ≅ G for each v ϵ S. This result, together with work of Fein and Schacher, yield the following. Let k be a global field of finite characteristic p, G a finite group. Then a necessary and sufficient condition that every division ring with center k and index equal to the order of G be a crossed product for G, is that k and G satisfy the hypothesis of the above theorem.