Isomorphism problems for Hopf-Galois structures on separable field extensions

TL;DR

This paper investigates the isomorphism conditions of Hopf algebras arising from Hopf-Galois structures on separable field extensions, providing criteria and classifications especially for cyclic extensions of prime power degree.

Contribution

It formulates criteria for isomorphism of Hopf algebras in Hopf-Galois structures and completes a detailed classification for cyclic extensions of degree p^n.

Findings

Criteria for Hopf algebra isomorphism established

Classification of Hopf-Galois structures on cyclic p^n extensions completed

Conditions for isomorphism as K-algebras in the commutative case determined

Abstract

Let $ L/K $ be a finite separable extension of fields whose Galois closure $ E/K $ has group $ G $. Greither and Pareigis have used Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on $ L/K $ has the form $ E[N]^{G} $ for some group $ N $ such that $ |N|=[L:K] $. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as $ K $-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and $ K $-algebras that appear in the classification of Hopf-Galois structures on a cyclic extension of degree $ p^{n} $, for $ p $ an odd prime number.