Is the ring $\mathbb{Z}\_p [[x]]\otimes\_{\mathbb{Z}\_p} \overline{\mathbb{Q}}\_p$ Noetherian?



Is the ring $\mathbb{Z}_p [[x]]\otimes_{\mathbb{Z}_p} \overline{\mathbb{Q}}_p$ Noetherian?
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.


