Isomorphism problems for Hopf–Galois structures on separable field extensions

Abstract

Let L/K be a finite separable extension of fields whose Galois closure E/K has Galois group G. Greither and Pareigis use 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 of order [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 pn, for p an odd prime number.