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 off-line 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 not-abelian extensions.