Does $[KL:L]=[K:K\cap L]$ when $L/(K\cap L)$ is normal?

Question

Let $K$ and $L$ be fields (inside a common ambient field) with $L/(K\cap L)$ normal. Is it always true that $[KL:L]=[K:K\cap L]$?

This is true when $K/(K\cap L)$ is a finite Galois extension, but I am having trouble producing a counterexample with $L/(K\cap L)$ is purely inseparable.

Answer 1

Here's a counterexample: Let $F=\mathbb{F}_2(t,u)$, let $\alpha$ be a root of $x^4+tx^2+u$, let $K=F(\alpha)$, let $L=F(t^{1/2},u^{1/2})$.

• $[L:F]=4$, since $x^4+tx^2+u$ is irreducible
• $[KL:L]=2$, since $x^4+tx^2+u$ factors as $(x^2+t^{1/2}x+u^{1/2})^2$ over $L$.
• $K\cap L=F$ by the answers to Does an inseparable extension have a purely inseparable element?