On the self-duality of rings of integers in tame and abelian extensions

Abstract

Let $L/K$ be a tame and Galois extension of number fields with group $G$. It is well-known that any ambiguous ideal in $L$ is locally free over ${\mathcal{O}}_{K}G$ (of rank one), and so it defines a class in the locally free class group of ${\mathcal{O}}_{K}G$, where ${\mathcal{O}}_K$ denotes the ring of integers of $K$. In this paper, we shall study the relationship among the classes arising from the ring of integers ${\mathcal{O}}_L$ of $L$, the inverse different ${\mathfrak{D}}_{L/K}^{-1}$ of $L/K$, and the square root of the inverse different $A_{L/K}$ of $L/K$ (if it exists), in the case that $G$ is abelian. They are naturally related because $A_{L/K}^2={\mathfrak{D}}_{L/K}^{-1}={\mathcal{O}}_L^{\ast }$, and $A_{L/K}$ is special because $A_{L/K}=A_{L/K}^{\ast }$, where $\ast$ denotes dual with respect to the trace of $L/K$.

Accepted version

References in corpus (1)

• On the Galois module structure of the square root of the inverse different in abelian extensions