Let $K$ be a (possibly infinite) field of characteristic $p$, and $L$ be a finite field extension of $\mathbb{F}_p$, so that $L$ is finite and $L/\mathbb{F}_p$ is Galois. Suppose $K \otimes_{\mathbb{F}_p} L$ is a field, can we conclude that $(K \otimes_{\mathbb{F}_p} L) / K$ is Galois? If not, what are some sufficient conditions for this conclusion to hold?
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1$\begingroup$ The tensor product of two fields will not in general be a field. $\endgroup$– Dmitry VaintrobCommented Apr 29, 2020 at 9:51
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$\begingroup$ Yes you are right. Let me make an edit. $\endgroup$– ZooradoCommented Apr 29, 2020 at 10:37
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2$\begingroup$ Yes. The action of the Galois group $G$ on $L$ extends to an action on $K\otimes _{\mathbb{F}_p}L$, and since the functor $K\otimes _{\mathbb{F}_p}-$ is exact, the $G$-invariant subfield is $K\otimes_{\mathbb{F}_p}\mathbb{F}_p=K$. Thus $(K\otimes_{\mathbb{F}_p}L)/K$ is Galois with group $G$. $\endgroup$– abxCommented Apr 29, 2020 at 12:17
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$\begingroup$ Your argument seems to not require any assumption of finiteness of the fields or of the extension. Is that correct? $\endgroup$– ZooradoCommented Apr 29, 2020 at 14:42
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