Automatic realization of Galois groups with cyclic quotient of order \(p^n\) (Q1011963)

This article provides a proof of a family of ``automatic realizability'' results for Galois extensions. An automatic realization of Galois groups is a pair of groups \((A,B)\) 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 \(F\). Such an automatic realization is nontrivial if the group \(B\) is not a quotient of the group \(A\). In this paper, the authors use equivariant Kummer theory to express certain automatic realization problems in terms of Galois modules, and then solve the corresponding Galois module problem. This is a substantially different approach from the more traditional analysis of embedding problems. This article specifically uses this approach to establish automatic realizations for a family of finite metacyclic \(p\)-groups. This family can be described as follows: For \(p\) a prime, \(n\) a positive integer, and \(G\) a cyclic group of order \(p^n\) with generator \(\sigma\), the group ring \({\mathbb{F}}_p[G]\) has \(p^n\) nonzero ring quotients \(M_j, 1 \leq j \leq p^n\), given by \(M_j := {\mathbb{F}}_p[G]/\langle (\sigma-1)^j\rangle\). Each \(M_j\) is a \(G\)-module since multiplication in the ring \({\mathbb{F}}_p[G]\) induces an \({\mathbb{F}}_p[G]\)-action on each \(M_j\). We let \(M_j\rtimes G\) denote the semidirect product. The following automatic realizability result is proved for groups over fields of characteristic \(p\) as well as fields of characteristic not \(p\). {Theorem:} For all \(0 \leq i < n\), realizability of any of the groups \(M_{k} \rtimes G, p^{i} + 1 \leq k < p^{i+1}\), implies realizability of the group \(M_{p^{i+1}}\rtimes G\).