# Thursday 17 Oct 2019: NT Seminar: Induced Hopf Galois structures and its Hopf Galois modules

### Prof. Anna Rio - Universitat Politecnica de Catalunya - UPC

**H103** 14:30-16:30

Hopf Galois theory is a generalization of Galois theory in which the algebraic object attached to a field extension L/K can be a Hopf algebra other than the group algebra K[G], or more concretely, a pair formed by a K−Hopf algebra H and a K−linear action µ of H over L which respects both the K−algebra structure of L and the K−Hopf algebra structure of H. This pair is what we call a Hopf Galois structure of L/K, and we say that L/K is a Hopf Galois extension when such a pair exists. With these definitions, Hopf Galois extensions may have several non-isomorphic Hopf Galois structures. In joint work with Crespo and Vela we introduced the notion of induced Hopf Galois structure, which is derived from smaller ones. We develop this definition by describing in more detail the attached Hopf algebra and the corresponding Hopf action. On the other hand, Hopf Galois theory can be used to broaden the domain of Galois module theory, giving rise to the so called Hopf Galois module theory. If L/K is an extension of p−adic fields, we are interested in the structure of its valuation ring OL as module over its associated order AH in H (that is, the maximal OK-order of H acting over L), for each Hopf Galois structure (H, µ) of L/K. Freeness of the valuation ring over the associated order is linked to the ramification and is of special interest the wild ramification case. In this talk we address the case of induced Hopf Galois structures and see how the results can be applied to dihedral degree 2p extensions of p−adic fields, where all the abelian non classical Hopf Galois structures are of induced type. In this frame we are able to relate freeness of such a dihedral extension to the known case of cyclic degree p extensions of p−adic fields