Galois Action and Localization in Number Fields

TL;DR

This work analyzes how the Galois group $G=\mathrm{Gal}(K/\mathbb{Q})$ acts on the class group $Cl_K$ of a Galois number field $K$, introducing a norm-like action framework that imposes strong structural constraints on $Cl_K$ and its automorphisms. By combining orbit-stabilizer arguments with a norm property, it derives divisibility and congruence conditions on class groups (e.g., $h_K\equiv 0$ or $1\pmod p$ for degree $p^r$) and rules out certain abelian groups as possible class groups in low-degree Galois extensions; it also studies the inverse class group problem and cycle-sum phenomena in $\mathrm{Aut}(Cl_K)$. The paper then shows how localization of rings of integers via $\mathcal O_K[1/x]$ preserves a natural Galois action on the localized class group and realizes any homomorphic image of $Cl_K$ as such a localization, yielding a Jordan–Hölder-like chain of overrings terminating in a PID. These results provide practical tools for understanding overrings of $\mathcal O_K$, factorization properties, and the interplay between Galois symmetry and class groups in number fields.

Abstract

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. While previous work on this problem has focused on representation theory, we take a direct approach to some classical and new problems. The paper concludes with an exploration of 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 $\mathcal{O}_K$.

Theorems & Definitions (32)

• Definition 1.1
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• proof
• Theorem 3.4
• proof
• Corollary 3.5
• proof