M ar 2 00 6 AUTOMATIC REALIZATIONS OF GALOIS GROUPS WITH CYCLIC QUOTIENT OF ORDER p n

We establish automatic realizations of Galois groups among groups M ⋊ G, where G is a cyclic group of order p for a prime p and M is a quotient of the group ring Fp[G]. The fundamental problem in inverse Galois theory is to determine, for a given field F and a given profinite group G, whether there exists a Galois extension K/F such that Gal(K/F ) is isomorphic to G. A natural sort of reduction theorem for this problem takes the form of a pair (A,B) of profinite groups with the property that, for all fields F , the existence of A as a Galois group over F implies the existence of B as a Galois group over F . We call such a pair an automatic realization of Galois groups and denote it A =⇒ B. The trivial automatic realizations are those given by quotients of Galois groups; by Galois theory, if G is realizable over F then so is every quotient H . It is a nontrivial fact, however, that there exist nontrivial automatic realizations. (See [J1, J2, J3] for a good overview of the theory of automatic realizations. Some interesting automatic realizations of groups of order 16 are obtained in [GS], and these and other automatic realizations of finite 2-groups are collected in [GSS]. For comprehensive treatments of related Galois embedding problems, see [JLY] and [Le].) The usual techniques for obtaining automatic realizations of Galois groups involve an analysis of Galois embedding problems. In this paper we offer a new approach based on the structure of natural Galois modules: we use equivariant Kummer theory to reformulate realization problems in terms of Galois modules, and then we solve Galois module problems. We take this approach in proving Theorem 1, which Date: March 24, 2006. 2000 Mathematics Subject Classification. 12F10. Research supported in part by NSERC grant R0370A01, and by a Distinguished Research Professorship at the University of Western Ontario. Research supported in part by National Security Agency grant MDA904-02-10061.