Parametrizing extensions with fixed Galois group

Abstract

We study the problem of parametrizing Galois extensions of fields with a fixed Galois group G. Similar problems have recently received much attention, as they are often a useful first step to finding the asymptotic number of such Galois extensions with bounded invariants. For example, Davenport and Heilbronn used a parametrization due to Levi of rings of rank three to count cubic field extensions of Q. Bhargava discovered and then used parametrizations of rings of ranks four and five to count quartic and quintic field extensions of Q. The approach taken by Levi and Bhargava can roughly be summarized as follows: Choose a basis of the field or ring extension R under consideration, and then write down the coefficients in the multiplication table of R with respect to this basis. To keep the multiplication table simple, one needs to choose a suitable basis of R, which has previously been accomplished in special cases using ad-hoc methods. In this thesis, we explain how the representation theory of G provides a convenient choice of basis in general.

This recovers many of the known parametrizations. We study cyclic groups G=Z/nZ in some detail and for example obtain a new parametrization for the cyclic group of order five. We also obtain a new parametrization for the quaternion group G={±1, ±i, ±j, ±k}. This allows us to count quaternionic extensions of Q by conductor.