Separability degree of extension as number of intermediate separable extensions.

Question

I read in another SE post that we can think of separability degree in this way but I can't explain why. Specifically, given an algebraic extension L/K, I want to understand this to see why the separability degree of L over the separable closure of K in L is 1.

Answer 1

Your question is a bit strange, for $E/F$ an algebraic extension $[E:F]_{sep}$ is defined as $[S:F]$ where $S$ is the set of elements of $E$ whose $F$-minimal polynomial is separable.

The main theorems are:

• that $S$ is a field,

• for $char(F)=0$ we have $S=E$,

• for $char(F)=p$ and $E=F(a_1,\ldots,a_n)$ we have $S=F(a_1^{p^m},\ldots,a_n^{p^m})$ whenever $p^m\ge [E:F]$,

• $[E:F]_{sep}$ is the number of $F$-homomorphisms $E\to \overline{E}$,

• therefore, for $M/E/F$ we have $[M:F]_{sep}=[M:E]_{sep}[E:F]_{sep}$.