Isomorphism problems for Hopf-Galois structures on separable field extensions

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.