TITLE: Is the ring $\mathbb{Z}_p [[x]]\otimes_{\mathbb{Z}_p} \overline{\mathbb{Q}}_p$ Noetherian?
QUESTION [5 upvotes]: Is the ring $\mathbb{Z}_p [[x]]\otimes_{\mathbb{Z}_p} \overline{\mathbb{Q}}_p$  Noetherian?

REPLY [11 votes]: Yes. Consider any nonzero element of the ring. Using the norm, we can see that it divides some nonzero element of $\mathbb Z_p [[x]] \otimes_{\mathbb Z_p} \mathbb Q_p$. By multiplication by $p$, it divides some nonzero element of $\mathbb Z_p[[x]]$. Applying Weierstrass preparation, we may factor it as a unit times a power of $p$ times a monic polynomial. If $f$ is a monic polynomial in $x$, then $\mathbb Z_p [[x]] \otimes_{\mathbb Z_p} \mathbb Q_p / f$ is finite over $\mathbb Q_p$, so tensoring it with $\overline{\mathbb Q}_p$ it will still be finite, hence Noetherian.
Since the quotient by any nonzero element is Noetherian, the ring is clearly Noetherian.