Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.

Question

This is a problem from Patrick Morandi's Field and Galois Theory: Chapter I.4, exercise 15.

Let $K$ be a finite extension of $F$. If $S=\{x\in K \mid x \text{ is separable over } F \}$ and $I=\{x\in K \mid x \text{ is purely inseparable over } F \}$ are the separable and purely inseparable closures of $F$ in $K$, respectively, we define the separable degree $[K:F]_s = [S:F]$ and the purely inseparable degree $[K:F]_i = [K:S]$.

Now using these definitions,

Prove the following product formulas for separability and inseparability degree: If $F \subseteq L \subseteq K$ are fields, then show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.

Proving just one of the equality will be enough (thanks to Tower law and the property that $[K:F]_s [K:F]_i = [K:F]$).

I started proving:

$[K:F]_s=[K:L]_s[L:F]_s$. Suppose $[K:L]_s=[S_1:F]$ and $[L:F]_s=[S_2:F]$ where $S_1$ and $S_2$ are separable closures of $K/L$ and $L/F$ respectively. And if we take $\{a_i \mid i=1,\dots,m\}$ and $\{b_i \mid i=1,\dots,n\}$ are the basis of $K/L$ and $L/F$ respectively. Is it true that $\{a_ib_j \mid i=1,\dots,m, j=1,\dots,n\}$ is a basis of $S/F$? Then for $x \in S$ how do we proceed?

Answer 1

Patrick Morandi sometimes poses exercises in a section earlier than you are expected to solve them. This seems to be one of those exercises. You can read Lemmas 8.9 and 8.11 on pages 82-83, which will help answer the problem thoroughly. I am giving their statements below:

Lemma 8.9. Let $K$ be a finite extension of $F$, and let $S$ be the separable closure of $F$ in $K$. Then $[S:F]$ is equal to the number of $F$-homomorphisms from $K$ to an algebraic closure of $F$.

Lemma 8.11. Suppose that $F \subseteq L \subseteq K$ are fields with $[K:F] < \infty$. Then $[K:F]_i = [K:L]_i [L:F]_i$.

To be honest, I find that the treatment in Lang's Algebra is far more intuitive, so do try to read the relevant sections from Lang to see how he discusses separable and purely inseparable extensions.

Prof. Lubin has shared his succint notes on this problem in the comments. The link is

www.math.brown.edu/~lubinj/sep.pdf

Another answer by Prof. Lubin (using Lang's definition of separable degree) is given here: $[K : F]_s = [K : L]_s [L : F]_s $ and $[K : F]_i = [K : L]_i [L : F]_i $