TITLE: Why does $E\otimes_KE\cong EG$ imply that Galois theory works?
QUESTION [18 upvotes]: 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." Ref: Algebra Vol 1: Fields and Galois Theory by Falko Lorenz.
Q: (solved by the comments below) What is the implication above? I thought $E\otimes_KE\cong EG$'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 $G\to \operatorname{Aut}_K(E)$? What is the author trying to express? Does this extend to infinite Galois extension case?
This question (Q) has been asked on Stackexchange but there is no answer. Hence, I asked this on MO. 
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.

REPLY [3 votes]: This is a CW answer to note that I answered this question over at math.SE. If someone upvotes this, it will be removed from the unanswered list.