Galoisgruppen lokaler Funktionenkörper
Zusammenfassung der Projektergebnisse
The computation of Galois groups of a given polynomial f is a classical problem of algorithmic number theory. About 40 years ago, first non-trivial algorithms have been developed by Stauduhar, McKay and Soicher for polynomials defined over the rationals. Based on those ideas the applicant and others generalised those ideas to compute Galois groups of polynomials (of arbitrary degree) over Q, Q(t), number fields, and global function fields. All these ground fields have in common that there is more or less easy access to an overfield, where the roots of the given polynomial can be at least approximatively represented. In case of the rational numbers this overfield can be the complex numbers or some suitably chosen p-adic field. When we would like to compute Galois groups over local fields, this approach fails for the reason that we do not find an easy overfield (except the splitting field) which contains the roots of f. For the case of p-adic fields Christian Greve developed in his PhD thesis new ideas for the computation of the Galois group. Using the idea of ramification polygons, he was able to write down a subfield T of splitting field N such that the Galois group of N/T is a p-group. In many situations he was able to describe the Galois group by generators and relations which very naturally appear from the ramification polygon structure. The idea of this project was to deal with the local function field case. On the one hand, we expected that almost all ideas of the p-adic case can be used as well. On the other hand, the local function field case should be easier than the p-adic case. Using the same methods as in the p-adic case, we can deal with at most tamely ramified extensions. Furthermore, we can analogously use the ideas of ramification polygons which allow us to write down an at most tamely ramified subfield of the splitting field. The new ingredient in the local function field case is that all local function fields are isomorphic to Laurent series rings over finite fields. This allows us to represent extension fields in this simpler representation. For this reason, a splitting field approach, which is usually not very practical, looks very promising in the local function field situation. These ideas will be described in the PhD thesis of Anthoula Zervou. All the described algorithms are implemented in Magma and there are a lot of examples, where the Galois groups have been computed. It is worth pointing out that there have been no previous implementations for computing the splitting field as well as the Galois group of a given polynomial over a local function field. Therefore, our algorithms are the first ones to compute them. Another part of the project was to extend the database for number fields which the applicant developed together with Gunter Malle. The number of included fields has increased, and we proved the minimal discriminants for many groups. We provided more statistical functions and added hyperlinks to databases describing (small) groups.
Projektbezogene Publikationen (Auswahl)
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Database for Number Fields
Jürgen Klüners
