Galois Action and Localization in Number Fields

For a Galois number field $K$, the Galois group $\text{Gal}(K/\mathbb{Q})$ acts on the class group $\text{Cl}_K$ in a very natural way: $\sigma\cdot[I]=[\sigma(I)]$ for any $\sigma \in \text{Gal}(K/\mathbb{Q})$, $[I]\in \text{Cl}_K$. In this paper, we will explore how the unique properties of this group action work together to elucidate the relationship between these two groups -- developing and expanding upon some known results from a new perspective. To this end, we explore the class groups of localizations of the ring of integers $\mathcal{O}_K$. These turn out to be powerful tools for understanding $\text{Cl}_K$ and overrings of $\mathcal{O}_K$. The paper concludes with some interesting observations about normset arithmetic and complexity -- topics intimately related to this action.