This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the galois group of $E/K$. $E[G]$ is the group ring formed by finite group $G$.

"It is worthwhile remarking that $E\otimes_KE\cong EG$ can be viewed as a deep reason why Galois theory works."

(Solved by the comments below.)**Q:** What is the implication above? I though $E\otimes_KE\cong EG$'s proof has a major ingredient that the trace map is non degenerate.(i.e $E/K$ is separable.) Is this affording some representation of $G\to Aut_K(E)$? What is the author trying to express? Does this extend to infinte galois extension case?

Ref: Algebra Vol 1: Fields and Galois Theory by Falko Lorenz

This question has been asked on stackexchange but there is no answer.(https://math.stackexchange.com/questions/2778581/why-e-otimes-ke-cong-eg-implies-that-galois-theory-works) Hence, I asked this on overflow.

$\textbf{Q':}$As qiaochu commented that this is related to torsor theory, I do not see why the two are related. It would be nice if someone could elaborate this a bit.

continuousfunctions ($G$ with the Krull topology and $K$ with the discrete topology). The title of his article is "The normal basis theorem for infinite extensions," or something close to that. $\endgroup$ – KConrad May 14 '18 at 18:59