Does normal Riemann hypothesis for $\zeta_{\mathbb{Q}}$ follows from the extended Riemann hypothesis for some $K \neq \mathbb{Q}$ (i.e. the statement that all zeroes of the Dedekind zeta function $\zeta_K$ for $K$ a number field in the critical strip lie on the axis $\mathfrak{R}(s)=1/2$)?
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Yes, certainly: zeta functions of abelian extensions of $\mathbb Q$ factor as products of Dirichlet $L$functions over $\mathbb Q$, including $\zeta(s)$, and those other $L$functions have no poles (which might cancel an offline zero of zeta), so RH for any abelian extension of $\mathbb Q$ implies that for $\mathbb Q$. At this point in history, the fact that we do not know that all Artin $L$functions lack poles in the critical strip severely complicates our provable understanding of an analogous assertion for notabelian extensions. 

