# A Note on Number Fields Sharing the List of Dedekind Zeta-Functions of Abelian Extensions with some Applications towards the Neukirch-Uchida Theorem

## TL;DR

This paper proves that the set of Dedekind zeta-functions of all finite abelian extensions uniquely determines a number field, providing a new approach to the Neukirch-Uchida theorem for non-normal extensions.

## Contribution

It establishes that the set of Dedekind zeta-functions characterizes a number field up to isomorphism, offering an alternative proof approach for Neukirch-Uchida theorem in certain cases.

## Key findings

- The set of Dedekind zeta-functions determines the number field.
- An alternative proof approach for Neukirch-Uchida theorem is proposed.
- The result applies to non-normal extensions of number fields.

## Abstract

Given a number field $K$ one associates to it the set $\Lambda_K$ of Dedekind zeta-functions of finite abelian extensions of $K$. In this short note we present a proof of the following Theorem: for any number field $K$ the set $\Lambda_K$ determines the isomorphism class of $K$. This means that if for any number field $K'$ the two sets $\Lambda_K$ and $\Lambda_{K'}$ coincide, then $K \simeq K'$. As a consequence of this fact we deduce an alternative approach towards the proof of Neukirch-Uchida Theorem for the case of non-normal extensions of number fields.