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Given a number field $K$ one associates to it the set $\Lambda_K$ of Dedekindzeta-functions of finite abelian extensions of $K$. In this short note wepresent 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 anynumber 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 approachtowards the proof of Neukirch-Uchida Theorem for the case of non-normalextensions of number fields.

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