Felipe Garcia Lopez (Heidelberg), An algorithm for computing Galois groups of additive polynomials.

Abstract:

Finite Galois extensions in positive characteristic are determined by the roots of separable additive polynomials, whose Galois groups occur as subgroups of suitable general linear groups over finite fields. The objective of the talk is to present an (implementable) algorithm for computing matrix group representations of the Galois groups of additive polynomials. The approach is based on the well-known Stauduhar method using so-called relative resolvents. We intend to compute the coefficients of these resolvents symbolically by specializing suitable invariants. A fast specialization process is presented in order to obtain practical applicability of the symbolic ansatz. Since we work in positive characteristic, the approach further uses linear versions of classical techniques for the computation of Galois groups. That is, we will see how to tailor the concept of the Tschirnhaus transformation to the context of additive polynomials, particularly to ensure termination of the algorithm. Moreover, we use ideas originally stemming from differential Galois theory to obtain a linear analogon of the classical Dedekind criterion.