My knowledge of Galois theory is woefully inadequate. Thus, I'd be interested in an exposition that assumes little knowledge of Galois theory, but is advanced in other respects. For instance, it would be nice if it were to include remarks like the following:
A finite field extension $K / k$ is separable iff the geometric fiber of Spec k -> Spec K is a finite union of reduced points.
[I was never able to remember what "separable" meant until I saw this equivalence while studying unramified morphisms. The proof is by the Chinese Remainder Theorem. Also note: this definition is incomplete, in the sense that it does not specify when a non-finite extension is separable.]
Is there any such exposition?
