p-Groups have unbounded realization multiplicity

2011/09/19 by Jen Berg, Berg, Jen, Andrew Schultz +1
Mathematics · #12F10 #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras #math.NT #msc:12F10

paper · pdf · doi:10.48550/arxiv.1109.4070

8 pages

arxiv created 2011/09/19 · openalex publication_date 2011/09/19 · arxiv updated 2011/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Parameterizing solutions to any Galois embedding problem over Z/pⁿZ with elementary p-abelian kernel
• Galois module structure of some elementary p-abelian extensions
• Galois p-groups and Galois modules
• Construction of maximal unramified p-extensions with prescribed Galois groups
• On realizability of p-groups as Galois groups