On the self-duality of rings of integers in tame and abelian extensions
Volume 191 / 2019
Abstract
Let $L/K$ be a tame Galois extension of number fields with group $G$. It is well-known that any ambiguous ideal in $L$ is locally free over $\mathcal {O}_KG$ (of rank one), and so it defines a class in the locally free class group of $\mathcal {O}_KG$, 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 $A_{L/K}^2 = \mathfrak {D}_{L/K}^{-1} = \mathcal {O}_L^*$, and $A_{L/K}$ is special because $A_{L/K} = A_{L/K}^*$, where $*$ denotes the dual with respect to the trace of $L/K$.
