Hopf-Galois Module Structure Of Some Tamely Ramified Extensions

Abstract

We study the Hopf-Galois module structure of algebraic integers in some finite extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is a finite unramified extension of $ p $-adic fields which is Hopf-Galois for some Hopf algebra $ H $ then the ring of algebraic integers $ \OL $ is a free module of rank one over the associated order $ \AH $. If $ H $ is a commutative Hopf algebra, we show that this conclusion remains valid in finite ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. We prove analogous results for finite abelian Galois extensions of number fields, in particular showing that if $ L/K $ is a finite abelian domestic extension which is Hopf-Galois for some commutative Hopf algebra $ H $ then $ \OL $ is locally free over $ \AH $. We study in greater detail tamely ramified Galois extensions of number fields with Galois group isomorphic to $ C_{p} \times C_{p} $, where $ p $ is a prime number. Byott has enumerated and described all the Hopf-Galois structures admitted by such an extension. We apply the results above to show that $ \OL $ is locally free over $ \AH $ in all of the Hopf-Galois structures, and derive necessary and sufficient conditions for $ \OL $ to be globally free over $ \AH $ in each of the Hopf-Galois structures. In the case $ p = 2 $ we consider the implications of taking $ K = \Q $. In the case that $ p $ is an odd prime we compare the structure of $ \OL $ as a module over $ \AH $ in the various Hopf-Galois structures.