Algorithmic Hopf Galois theory

Abstract

[en] Chase and Sweedler introduce Hopf Galois theory, which is a generalization of Galois theory. The point is to replace the Galois group by a Hopf algebra and the Galois action (by automorphisms) by an action by endomorphisms called Hopf action. This pair gives the so-called Hopf Galois structure. In the case of separable field extensions Greither and Pareigis characterize Hopf Galois structures in terms of groups. This characterization gives a method to determine all Hopf Galois structures of a given separable extension. In this thesis we present two algorithms written in the computational algebra system Magma to compute all Hopf Galois structures of a given separable extension. Moreover they determine two important properties of the computed Hopf Galois structures. The first algorithm is based on Greither-Pareigis’ theorem. It is very efficient but it just reaches degree 11. In order to go further, we develop the second algorithm, which is based on Byott’s translation theorem. Therefore in this memory we also detail the proofs of both theorems.