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2 corrected two typos

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 the $K$ is finitely generated as a field over $K$. 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.

1

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 the $K$ is finitely generated as a field over $K$.

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.
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