Abstract: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.
From: Jared Kettinger [view email]
[v1]
Sat, 11 Oct 2025 04:47:41 UTC (38 KB)
[v2]
Sun, 15 Feb 2026 19:29:12 UTC (40 KB)
[v3]
Wed, 4 Mar 2026 10:37:18 UTC (48 KB)
[v4]
Tue, 10 Mar 2026 04:39:20 UTC (50 KB)
[v5]
Tue, 2 Jun 2026 17:16:26 UTC (51 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。