Intermediate fields in extension

Question

I don't know how to prove this inequality from Hungerford's Algebra book:

Let $L$ and $M$ be intermediate fields in the extension $K\subset F$. Prove that $$[LM:K]\le[L:K][M:K].$$ Here, $LM$ is the subfield generated by $L\cup M.$

Answer 1

If $L$ and $M$ are algebraic extensions, $LM$ is generated as a $k$-vector space by products of generators of $L$ and of $M$.

EDIT: I've worked out the infinite case, if you want it. The main observation is that a transcendental extension of $K$ has degree at least the cardinality of $K$, since if $x$ is transcendental then the elements $1/(x-a)$ are linearly independent as $a$ runs over $K$. Thus such an extension $M$ has degree equal to its own cardinality: its cardinality is the cardinal product $|K|[M:K]=[M:K]$. Now $[LM:K]=|LM|=\max(|L|,|M|)=\max([L:K],[M:K])=[L:K][M:K]$.