Bi-Quadratic Number Fields with Trivial 2-Primary Hilbert Kernels

Abstract

Using results of Browkin and Schinzel one can easily determine quadratic number fields with trivial 2-primary Hilbert kernels. In the present paper we completely determine all bi-quadratic number fields which have trivial 2-primary Hilbert kernels. To obtain our results, we use several different tools, amongst which is the genus formula for the Hilbert kernel of an arbitrary relative quadratic extension, which is of independent interest. For some cases of real bi-quadratic fields there is an ambiguity in the genus formula, so in this situation we use instead Brauer relations between the Dedekind zeta-funtions and the Birch–Tate conjecture. 2000 Mathematics Subject Classification 11R70, 19F15.