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.

  • 4
    $\begingroup$ 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 $\text{Aut}(E/K)$. $\endgroup$
    – KConrad
    May 14, 2018 at 18:44
  • $\begingroup$ @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? $\endgroup$
    – user45765
    May 14, 2018 at 18:51
  • 4
    $\begingroup$ 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. $\endgroup$
    – KConrad
    May 14, 2018 at 18:59
  • 1
    $\begingroup$ The assertion is equivalent to the normal basis theorem which some books do use to give the fundamental theorem of Galois theory. $\endgroup$ May 14, 2018 at 19:56
  • 1
    $\begingroup$ 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. $\endgroup$
    – F Zaldivar
    May 15, 2018 at 14:52

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.