Constructive Galois theory
In constructive Galois theory, there are two main questions: the direct problem and the inverse problem. For the inverse problem the question is whether it is possible to find a polynomial such that the Galois group of that polynomial is a given finite group. In this talk, we will focus on the direct problem. Given a rational polynomial f, we explain how to compute the Galois group of this polynomial. The presented algorithms are implemented in Magma and work without any degree restriction. These methods apply over number fields and global function fields, too.
In a second part of the talk, we report on a database containing number fields up to degree 23. The database is complete in the sense that for each transitive group (with two exceptions) up to degree 23 there is at least one polynomial in the database which has this given Galois group. The database can be accessed via galoisdb.math.uni-paderborn.de.