# Why does $E\otimes_KE\cong EG$ imply that Galois theory works?

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.

• If $E/K$ is finite then $|\text{Aut}(E/K)| \leq [E:K]$, with equality if and only if $E/K$ is a Galois extension. With this in mind, compute the $E$-dimension of both sides, letting $G$ be $\text{Aut}(E/K)$. – KConrad May 14 '18 at 18:44
• @KConrad I should deduce that $dim_E(E\otimes_KE)\geq dim_E(E[G])$ as $Aut(E/K)\leq[E:K]$. The content of the $E\otimes_KE\cong EG$ is equivalent to $E/K$ is galois for finite extension case. Is this correct? I also know $k[G]\cong E$ as $k[G]$ modules though non-canonical. So for finite extension, $k[G]\cong E$ iff $E[G]\cong E\otimes_KE$ iff $E/K$ galois. Is this interpretation correct? And what can I conclude about infinite extension via inverse limit of galois group? Can I pull out the inverse limit as direct limit on the other side? – user45765 May 14 '18 at 18:51
• Hendrik Lenstra has a result that generalizes the isomorphism $K[G]\cong E$ in the infinite-degree Galois case: think of the group ring as the functions $G \rightarrow K$. When $G$ is infinite, work with continuous functions ($G$ 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. – KConrad May 14 '18 at 18:59
• The assertion is equivalent to the normal basis theorem which some books do use to give the fundamental theorem of Galois theory. – Benjamin Steinberg May 14 '18 at 19:56
• For the relation to torsors in your second question, the point is that the normal basis theorem could be seen as a consequence of Galois descent for algebras. – F Zaldivar May 15 '18 at 14:52