On Exceptions in the Brauer–Kuroda Relations
Volume 59 / 2011
Bulletin Polish Acad. Sci. Math. 59 (2011), 207-214
MSC: Primary 20D15; Secondary 11R42.
DOI: 10.4064/ba59-3-3
Abstract
Let $F$ be a Galois extension of a number field $k$ with the Galois group $G$. The Brauer–Kuroda theorem gives an expression of the Dedekind zeta function of the field $F$ as a product of 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.
