Comparing degrees of field extensions: compositum over field versus field over intersection.

Question

I'm stuck on the following question:

For fields $L$ and $K$, show that it is possible that $[LK:K]<[L:L\cap K]$.

I have realized that it is not possible if $L$ is an extension of $K$. This is because the degree of $LK$ over $K$ would be the same as the degree of $L$ over $K$, which is the same as the degree of $L$ over $L\cap K=K$. I recognize that if $L$ and $K$ are groups, then it is an equality by second isomorphism theorem (I think? that applies to subgroups of the same group) but I simply cannot get any farther on this. Any ideas?

Answer 1

Let $\alpha$ satisfy an irreducible cubic $f$ over $\mathbb{Q}$, such that $\mathbb{Q}(\alpha)$ is not Galois over $\mathbb{Q}$.

Let $\alpha'$ be another root of $f$.

Consider $L := \mathbb{Q}(\alpha)$, $K := \mathbb{Q}(\alpha')$. Their intersection is $\mathbb{Q}$. I'll leave it up to you to figure out their compositum.