UC Santa Barbara

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.