Galois cohomology of ambiguous ideals

Abstract

$\text{K}\text{F}$ denotes a finite Galois extension with Galois group G, F the quotient field of a Dedekind domain with finite residue class fields. We characterize the cohomologically trivial ambiguous ideals of K. When $\text{K}\text{F}$ is cyclic, the cohomology groups of an ambiguous ideal of K have the same order. This is shown to fail in the noncyclic case.