  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
$\mathcal{O}_KG$ (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 dual with respect to the trace of
$L/K$.
