Suppose that $k$ is an *algebraically closed* field. Let $F/k$ be a (possibly non-finitely generated) field extension. Is


$$
k[[x]] \otimes\_{k} F
$$


noetherian?


If not, is the natural map $k[[x]] \otimes\_{k} F \to F[[x]]$ injective?



Is the tensor product of a power series ring and a field noetherian?
The answer is no: for example, $k[[x]]\otimes\_k k((x))$ is not noetherian.  

Indeed, if it were, so would be $ k((x))\otimes\_k k((x))$.  

But this would contradict the following interesting general theorem of Vámos: 


Given an extension of fields $K/F$ the tensor product $K\otimes\_F K$ is noetherian if and only if $K$ is finitely generated as a field over $F$. 


**Full confession**  

 I have only read an abstract of Vámos's article because I have no access to it. Anyway, here is the reference:  

P. Vámos, On the minimal prime ideal of a tensor product of two fields, Math. Proc.
Cambridge Philos. Soc. 84 (1978), no. 1, p.25-35.


