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

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 $K^{\prime }$ the two sets ${\Lambda }_K$ and ${\Lambda }_{K^{\prime }}$ coincide, then $K\simeq K^{\prime }$. 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.

References in corpus (2)

• On Artin L-functions and Gassmann Equivalence for Global Function Fields
• Reconstructing global fields from Dirichlet L-series