Galois action on a tensor product of fields Let $K$ be a field, $F$ a finite field extension of $K$ and let $L$ be an algebraic closure of $K$. Let $G_K:=\operatorname{Gal}(K^{\text{sep}}|K)$ be the absolute Galois group of $K$. $G_K$ acts on $L$. We have
$$
F\otimes_K L=\prod_{\nu}{L}
$$
where the product is indexed over field embeddings $\nu:F\to L$. Any $\sigma\in G_K$ acts on the left-hand side of this equality by $1\otimes \sigma$. What is the corresponding action on the right-hand side in terms of Galois groups?
Many thanks!
 A: I think you assume that $F/K$ is separable, right ? For the tensor product to be reduced. Ok, so in order to obtain the induced action of $G_K$ on the right hand side, one needs first to fix a $K$-isomorphism that identifies the left hand side with the right hand side.
Writing $F$ as $K[X]/(f)$ for an irreducible polynomial $f$ (the minimal ideal of a primitve element), we can write the left hand side as $L[X]/(f)$ with the obvious $G_K$ action on it. Now take a complete factorization of $$f=(X-a_1)\cdots(X-a_n)$$ in $L[X]$ and apply the Chinese remainder theorem to obtain the canonical isomorphism
\begin{align*}
L[X]/(f) &\longrightarrow L[X]/(X-a_1)  \times \ldots \times  L[X]/(X-a_n)\\  
g+(f) &\mapsto  (g + (X-a_1), ..., g + (X-a_n) ) 
\end{align*}
On the righthand side you have the cartesian product that you mentioned. If you want to know the induce $G_K$ action, you have to start with an arbitrary tuple on the right and side, pull it back to the left side, apply an element of $G_K$ and push the result back to the right hand side.
You will see that what you obtain is a superposition of a transitive permutation action of the components (corresponding to how the elements a_1,...,a_n are permuted) and a straight forward action of the elements of each component.
Sorry for not writing this up with more details provided. I just came across your question when I was looking for mentioning of a reference in the literature of this very observation.
Thus let me extend your question (hoping that maybe you have in the meantime the answer to my extended question): Does anybody know a reference in the literature where not only the equality of the tensor product and the cartesian product as k-algebras is stated, but also the induced $G_K$ action on the cartesian product is described ?
