Let K be a field and let N be a finitely generated group with finite automorphism group F. As shown by Haggenmuller and Pareigis, there is a bijection Θ from the collection of F-Galois extensions of K to the collection of forms of the Hopf algebra K[N]. In the case that K is a finite field extension of Q and H is the Hopf algebra of a Hopf-Galois structure on a Galois extension E/K, we construct the preimage of H under Θ. We give criteria to determine the Hopf algebra isomorphism classes of the Hopf algebras attached to the Hopf-Galois structures on E/K. Examples are included throughout the paper.

The authors are indebted to the referee whose thoughtful comments and suggestions have improved this paper.