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

Let $K$ be a number field with ring of integers $\\mathcal{O}_K$ and let $G$ be a finite \\mbox{group of odd} order. Given a $G$-Galois $K$-algebra $K_h$, let $A_h$ be the fractional ideal in $K_h$ whose square is the inverse different of $K_h/K$, which exists by Hilbert's formula since $G$ has odd order. By a theorem of B. Erez, we know that $A_h$ is locally free over $\\mathcal{O}_KG$ when $K_h/K$ is \\emph{weakly ramified}, i.e. all of the second ramification groups in lower numbering attached to $K_h/K$ are trivial. In this case, the module $A_h$ determines a class $\\mbox{cl}(A_h)$ in the locally free class group $\\mbox{Cl}(\\mathcal{O}_KG)$ of $\\mathcal{O}_KG$. Such a class in $\\mbox{Cl}(\\mathcal{O}_KG)$ will be called \\emph{$A$-realizable}, and \\emph{tame $A$-realizable} if $K_h/K$ is in fact tame. We will write $\\mathcal{A}(\\mathcal{O}_KG)$ and $\\mathcal{A}^t(\\mathcal{O}_KG)$ for the sets of all $A$-realizable classes and tame $A$-realizable classes in $\\mbox{Cl}(\\mathcal{O}_KG)$, respectively.In this dissertation, we will consider the case when $G$ is abelian. First of all, we will show that $\\mathcal{A}^t(\\mathcal{O}_KG)$ is in fact a subgroup of $\\mbox{Cl}(\\mathcal{O}_KG)$ and that a class $\\mbox{cl}(A_h)\\in\\mathcal{A}(\\mathcal{O}_KG)$ is tame $A$-realizable if the wildly ramified primes of $K_h/K$ satisfy suitable assumptions. Our result will imply that $\\mathcal{A}(\\mathcal{O}_KG)=\\mathcal{A}^t(\\mathcal{O}_KG)$ holds if the primes dividing $|G|$ are totally split in $K/\\mathbb{Q}$. Then, we will show that $\\Psi(\\mathcal{A}(\\mathcal{O}_KG))=\\Psi(\\mathcal{A}^t(\\mathcal{O}_KG))$ holds without any extra assumptions. Here $\\Psi$ is the natural \\mbox{homomorphism $\\mbox{Cl}(\\mathcal{O}_KG)\\longrightarrow\\mbox{Cl}(\\mathcal{M}(KG))$} afforded by extension of scalars and $\\mbox{Cl}(\\mathcal{M}(KG))$ denotes the locally free class group of the maximal $\\mathcal{O}_K$-order $\\mathcal{M}(KG)$ in $KG$. Last but not least, we will show that the group structure of $\\mathcal{A}^t(\\mathcal{O}_KG)$ is connected to the study of embedding problems.