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?