Galois Action and Localization in Number Fields

For a Galois number field $K$, the Galois group $\text{Gal}(K\mathrm{/}\mathbb{Q})$ acts on the class group ${\text{Cl}}_K$ in a very natural way: $\sigma \cdot [I]=[\sigma (I)]$ for any $\sigma \in \text{Gal}(K\mathrm{/}\mathbb{Q})$, $[I]\in {\text{Cl}}_K$. 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.