Isomorphisms of Galois groups of number fields with restricted ramification

TL;DR

This paper investigates whether isomorphisms between Galois groups of number fields with restricted ramification imply unique isomorphisms between the fields themselves, under certain density assumptions on the ramification sets.

Contribution

It provides conditions under which Galois group isomorphisms uniquely determine the underlying number fields with restricted ramification.

Findings

Galois group isomorphisms correspond to field isomorphisms under density conditions

Unique field reconstruction from Galois groups is established for large ramification sets

Results depend on Dirichlet density assumptions of ramification primes

Abstract

Let $K$ be a number field and $S$ a set of primes of $K$. We write $K_S/K$ for the maximal extension of $K$ unramified outside $S$ and $G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions: "For $i=1,2$, let $K_i$ be a number field, $S_i$ a (sufficiently large) set of primes of $K_i$ and $\sigma :G_{K_1,S_1}\to G_{K_2,S_2}$ an isomorphism. Is $\sigma$ induced by a unique isomorphism between $K_{1,S_1}/K_1$ and $K_{2,S_2}/K_2$?" Here the main assumption is about the Dirichlet density of $S_i$.