Proposition 15

Let E, be the relative separable closure of K in E; then $[E : K]_s = [E_s : K]$.

The field $\Omega$ is perfect and E is p-radical over $E_s$, by V, p. 44, Prop. 13; therefore Prop. 3 (V, p. 26) shows that every K-homomorphism of E, into $\Omega$ extends in a unique fashion to a K-homomorphism of E into $\Omega$; we thus have $[E : K]_s = [E_s : K]$. Since E, is a separable extension of finite degree of K, it is an etale algebra over K; so we have $[E_s : K]_s = [E_s : K]$ by V, p. 32, Prop. 4, and the result follows.

With the preceding notation, the degree of E over $E_s$ is called the inseparable degree of E over K and is denoted by $[E : K]_i$. We thus have

(1)

by Prop. 15.

When K is of characteristic 0, then $E = E_s$, and so $[E : K]_s = [E : K]$ and $[E : K]_i = 1$. If K is of characteristic $p \neq 0$, the number $[E : K]_i$ is a power of p because E is p-radical over $E_s$ (V, p. 44, Prop. 13 and p. 26, Prop. 4). It should be noted that $[E : K]_i$ is not necessarily equal to the highest power of $p$ dividing $[E : K]_s$, nor equal to the degree $[E_s : K]$ of the relativep-radical closure of E in K (V, p. 152, Ex. 3 and 2).