Galois Action and Localization in Number Fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912956983279616 |
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| author | Coykendall, Jim Kettinger, Jared |
| author_facet | Coykendall, Jim Kettinger, Jared |
| contents | For a Galois number field $K$, the Galois group $\text{Gal}(K/\mathbb{Q})$ acts on the class group $Cl_K$ in a very natural way: $σ\cdot[I]=[σ(I)]$ for any $σ\in \text{Gal}(K/\mathbb{Q})$, $[I]\in Cl_K$. 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 $Cl_K$ and overrings of $\mathcal{O}_K$. The paper concludes with some interesting observations about normset arithmetic -- a topic intimately related to this action. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10018 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Galois Action and Localization in Number Fields Coykendall, Jim Kettinger, Jared Number Theory Commutative Algebra 11R04 For a Galois number field $K$, the Galois group $\text{Gal}(K/\mathbb{Q})$ acts on the class group $Cl_K$ in a very natural way: $σ\cdot[I]=[σ(I)]$ for any $σ\in \text{Gal}(K/\mathbb{Q})$, $[I]\in Cl_K$. 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 $Cl_K$ and overrings of $\mathcal{O}_K$. The paper concludes with some interesting observations about normset arithmetic -- a topic intimately related to this action. |
| title | Galois Action and Localization in Number Fields |
| topic | Number Theory Commutative Algebra 11R04 |
| url | https://arxiv.org/abs/2510.10018 |