Abelian Hopf Galois structures from almost trivial commutative nilpotent algebras

zbMATH Open1435.12003arXiv1604.05269MaRDI QIDQ2280167

Lindsay N. Childs

Publication date: 18 December 2019

Published in: The New York Journal of Mathematics (Search for Journal in Brave)

Abstract: Let $L/K$ be a Galois extension of fields with Galois group $G$, an elementary abelian $p$-group of rank $n$ for $p$ an odd prime. It is known that nilpotent $mathbb{F}_p$-algebra structures $A$ on $G$ yield regular subgroups of the holomorph of $G$, hence Hopf Galois structures on $L/K$. In this paper we illustrate the richness of Hopf Galois structures on $L/K$ by examining the case where $A$ is abelian of dimension $n$ where the dimension of $A^2=1$. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent of failure of surjectivity of the Galois correspondence for those structures.

Full work available at URL: https://arxiv.org/abs/1604.05269

File on IPFS (Hint: this is only the Hash - if you get a timeout, this file is not available on our server.)

zbMATH Keywords

finite commutative nilpotent algebrasabelian Hopf Galois extensions of fieldsregular subgroups of finite affine groups

Mathematics Subject Classification ID

Separable extensions, Galois theory (12F10) Subgroups of symmetric groups (20B35) Structure of finite commutative rings (13M05) Galois theory and commutative ring extensions (13B05)

Related Items (2)

Some Hopf Galois structures arising from elementary abelian $p$-groups ⋮ Bounds on the number of ideals in finite commutative nilpotent $\mathbb{F}_p$-algebras

This page was built for publication: Abelian Hopf Galois structures from almost trivial commutative nilpotent algebras

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q2280167)