# Why $E\otimes_KE\cong EG$ implies 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."

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?

• Which book is this from ? – Rene Schipperus May 13 '18 at 0:00
• @ReneSchipperus Falko. Algebra Vol I:Fields and Galois Theory – user45765 May 13 '18 at 0:02
• It's a long story. The keyword you want to look up is "torsor." – Qiaochu Yuan May 13 '18 at 2:15
• @QiaochuYuan Is there a book that explains this story in detail? – Rene Schipperus May 13 '18 at 2:26
• Now posted to MO, mathoverflow.net/questions/300197/… – Gerry Myerson May 14 '18 at 23:06