Algorithms for Galois extensions of global function fields

Abstract

In this thesis we consider the computation of integral closures in cyclic Galois extensions of global function fields and the determination of Galois groups of polynomials over global function fields. The development of methods to efficiently compute integral closures and Galois groups are listed as two of the four most important tasks of number theory considered by Zassenhaus. We describe an algorithm each for computing integral closures specifically for Kummer, Artin--Schreier and Artin--Schreier--Witt extensions. These algorithms are more efficient than previous algorithms because they compute a global (pseudo) basis for such orders, in most cases without using a normal form computation. For Artin--Schreier--Witt extensions where the normal form computation may be necessary we attempt to minimise the number of pseudo generators which are input to the normal form. These integral closure algorithms for cyclic extensions can lead to constructing Goppa codes, which can correct a large proportion of errors, more efficiently. The general algorithm we describe to compute Galois groups is an extension of the algorithm of Fieker and Klueners to polynomials over function fields of characteristic p. This algorithm has no restrictions on the degrees of the polynomials it can compute Galois groups for. Previous algorithms have been restricted to polynomials of degree at most 23. Characteristic 2 presents additional challenges as we need to adjust our use of invariants because some invariants do not work in characteristic 2 as they do in other characteristics. We also describe how this algorithm can be used to compute Galois groups of reducible polynomials, including those over function fields of characteristic p. All of the algorithms described in this thesis have been implemented by the author in the Magma Computer Algebra System and perform effectively as is shown by a number of examples and a collection of timings.