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

Given a number field $K$ one associates to it the set ${\mathrm{\Lambda }}_K$ 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 ${\mathrm{\Lambda }}_K$ determines the isomorphism class of $K$. This means that if for any number field $K^{\mathrm{\prime }}$ the two sets ${\mathrm{\Lambda }}_K$ and ${\mathrm{\Lambda }}_{K^{\mathrm{\prime }}}$ 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.