Asymptotics of number fields and the Cohen--Lenstra heuristics

We study the asymptotics conjecture of Malle for dihedral groups $D_\ell$ of order $2\ell$, where $\ell$ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen--Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds

Computation of Galois groups of rational polynomials

Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial is a very old problem. Currently, the best algorithmic solution is Stauduhar's method. Computationally, one of the key challenges in the application of Stauduhar's method is to find, for a given pair of groups H<G a G-relative H-invariant, that is a multivariate polynomial F that is H-invariant, but not G-invariant. While generic, theoretical methods are known to find such F, in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.Comment: Improved version and new titl

A counter example to Malle's conjecture on the asymptotics of discriminants

In this note we give a counter example to a conjecture of Malle which predicts the asymptotic behaviour of the counting functions for field extensions with given Galois group and bounded discriminant