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

Let $L\mathrm{/}K$ 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 ${\mathfrak{D}}_{L\mathrm{/}K}^{-1}$ of $L\mathrm{/}K$, and the square root of the inverse different $A_{L\mathrm{/}K}$ of $L\mathrm{/}K$ (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}$isspecialbecause$A{L/K} = A_{L/K}^$,where$$denotesdualwithrespecttothetraceof$L/K$.