In the proof of Proposition 5.6.1 (the homotopy exact sequence) in Szamuely's book Galois Groups and Fundamental Groups, I have the following doubt. The proposition is as follows.
Proposition 5.6.1: Let $X$ be a quasi compact and geometrically integral scheme over a field $k$. Fix an algebraic closure $\overline{k}$ of $k$, and let $k_s/k$ be the corresponding separable closure. Write $\overline{X}:=X\times _{\operatorname{Spec} k} \operatorname{Spec}k_s$, and let $\overline{x}$ be a geometric point of $\overline{X}$ with values in $\overline{k}$. The sequence of profinite groups $$ 1\to \pi_1(\overline{X},\overline{x})\to \pi_1(X,\overline{x})\to \text{Gal } (k_s/k) \to 1$$ induced by the maps $\overline{X}\to X$ and $X\to \operatorname{Spec} k$ is exact.
Additionally, one requires in the proof that the scheme $X$ be quasi separated, although this hypothesis has been erroneously omitted in the book.
Now, towards the end of the proof, we have a Galois extension $K$ of the function field of $X$, $k(X)$, which splits into a direct product of copies of $k(\overline{X})=: k_s(X)$ when tensored with $k_s$. The author claims that this is the case if and only if $K$ itself is some $k(X)\otimes_k L$, where $L/k$ is a finite Galois extension, but I don't see why this is immediate. I would be grateful if someone could clarify why this is the case.
Thanks.