About Hopf-Galois structures in magma computation

Abstract

[en] Group theory is quite an astonishing field of Mathematics that reminds of a big world of puzzles, starting from the very first definition of a Group to the concept of the Monster Group which is featured in many informational papers and videos. In particular, one of the most interesting approaches is the Galois theory, which was first introduced in the bachelor’s degree. We will merge this concept together with algebras to achieve the structures that this work’s title is based on, the Hopf Galois structures. These structures will be the focal point of the present thesis. The goal is to compute them using Magma (short for Magma Computational Algebra System), a software designed for computations in algebra. For that matter, we will start by presenting the preliminaries where we give concepts that might have not been shown in the bachelor’s degree. Afterwards we will give the definitions of the types of algebra to be used. It is followed by the Greither-Pareigis theory which gives the background of the Hopf Galois structures. Sought after we have the Byott’s Theorem, which has an immediate application that our program will be based on. Finally, to summarize, we will show some other results and continuations about this matter.