p-Groups have unbounded realization multiplicity

TL;DR

This paper investigates the realization multiplicity of p-groups in Galois theory, showing that for certain classes of p-groups, this multiplicity can be arbitrarily large, by analyzing Galois embedding problems and module structures.

Contribution

It introduces a new approach linking Galois embedding problems with module theory to demonstrate unbounded realization multiplicity of p-groups.

Findings

Certain p-groups have unbounded realization multiplicity

Galois embedding problems can be interpreted via Galois submodules

Module structure analysis reveals classes of p-groups with infinite realization multiplicity

Abstract

In this paper we interpret the solutions to a particular Galois embedding problem over an extension K/F whose Galois group is a finite, cyclic p group in terms of certain Galois submodules within the parameterizing space of elementary p-abelian extensions of K; here p is a prime. Combined with some basic facts about the module structure of this parameterizing space, this allows us to exhibit a class of p-groups whose realization multiplicity is unbounded.