# On Locally GCD Equivalent Number Fields

## TL;DR

This paper discusses the rigidity of low-degree number field extensions under local GCD equivalence, showing they are uniquely characterized by prime splitting behavior, with a new proof emphasizing prime densities and basic Galois theory.

## Contribution

Provides a new proof of Lochter's rigidity result for low-degree extensions using prime densities and elementary Galois theory techniques.

## Key findings

- Number fields of degree ≤ 5 are uniquely determined by local GCD equivalence.
- Extensions of degree 3 and 5 are characterized by inert primes.
- The new proof relies on Chebotarev's Theorem and basic Galois theory.

## Abstract

Local GCD Equivalence is a relation between extensions of number fields which is weaker than the classical arithmetic equivalence. It was originally studied by Lochter with Weak Kronecker Equivalence. Among the many results he got, Lochter discovered that number fields extensions of degree $\leq 5$ which are locally GCD equivalent are in fact isomorphic. This fact can be restated saying that number fields extensions of low degree are uniquely characterized by the splitting behaviour of a restricted set of primes: in particular, also extensions of degree 3 and 5 are uniquely determined by their inert primes, just like the quadratic fields. The goal of this note is to present this rigidity result with a different proof, which insists especially on the densities of sets of prime ideals and their use in the classification of number fields up to isomorphism. Alongside Chebotarev's Theorem, no harder tools than basic Group and Galois Theory are required.