The Structure of Hopf Algebras Acting on Dihedral Extensions
We discuss isomorphism questions concerning the Hopf algebras that yield Hopf–Galois structures for a fixed separable field extension L/K. We study in detail the case where L/K is Galois with dihedral group \(D_p\), \(p\ge 3\) prime and give explicit descriptions of the Hopf algebras which act on L/K. We also determine when two such Hopf algebras are isomorphic, either as Hopf algebras or as algebras. For the case \(p=3\) and a chosen L/K, we give the Wedderburn–Artin decompositions of the Hopf algebras.
KeywordsHopf–Galois extension Dihedral extension Wedderburn–Artin decomposition
The authors would like to thank the referee for comments and suggestions which improved the exposition and content of this paper.
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