Criteria when $[ML:K]= [M:K] \cdot [L:K]$ holds

Question

Let $K$ be a field and $M,L$ algebraic field extensions and $ML$ the composition/product field.

I'm looking for some useful criteria when the equation

$$[ML:K]= [M:K] \cdot [L:K]$$

holds.

Preferably more structural orienting on special cases of $M$ and $L$ and deeper then for example when $[M:K],[L:K]$ are coprime.

Answer 1

Here's a fairly general result, not sure if it will be helpful.

Consider the canonical map $\phi : M\otimes_K L \to ML$ given by $m\otimes l \mapsto ml$. Observe that $\phi$ is surjective, and $$\dim_K M\otimes_K L = [M:K][L:K],$$ so $[ML:K]\le [M:K][L:K]$ with equality exactly when $\phi$ is an isomorphism (which is equivalent to $\phi$ being injective).

Note that if $M\otimes_K L$ is a field, then $\phi$ is injective, and conversely if $\phi$ is injective, then $M\otimes_K L\cong ML$ is a field.

Thus $[ML:K]=[M:K][L:K]$ exactly when $M\otimes_K L$ is a field.

This question, when is the tensor product of fields again a field, has been asked before on MO, and has received some good answers.

Lastly, I should note Wikipedia has an article on tensor products of fields, and the property that $M\otimes_K L \to ML$ is injective is called linear disjointness, and we say that $M$ and $L$ are linearly disjoint over $K$. See the link to the Wiki article for more on that.