  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.
