On Exceptions in the Brauer-Kuroda Relations

Journal article, 2011

Let F be a Galois extension of a number field k with Galois group G. The Brauer-Kuroda theorem gives an expression of the Dedekind zeta function of the field F as a product of the zeta functions of some of its subfields containing k, provided the group G is not exceptional. In this paper, we investigate the exceptional groups. In particular, we determine all nilpotent exceptional groups and give a sufficient condition for a group to be exceptional. We give many examples of nonnilpotent solvable and nonsolvable exceptional groups.