Galois Action and Localization in Number Fields

2025/10/11 by Coykendall, Jim, Kettinger, Jared
#11R04 #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.10018

Abstract

For a Galois number field K, the Galois group Gal(K/ℚ) acts on the class group Cl_K in a very natural way: σ⋅[I]=[σ(I)] for any σ∈ Gal(K/ℚ), [I]∈ 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 O_K. These turn out to be powerful tools for understanding Cl_K and overrings of O_K.

Citations

• The geometry of intersecting codes and applications to additive combinatorics and factorization theory
• On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms
• Galois action on class groups

Related

• Actions of Galois groups on invariants of number fields
• Galois representations
• Galois action on class groups
• On the localization map in the Galois cohomology of algebraic groups
• Vanishing of the KNil groups: localization methods