M:K and L:K are normal and finite extensions then M intersection L : K is also finite and normal.

Question

How to prove or disprove the following statement:

M:K and L:K are normal and finite extensions -> M intersection L : K is also finite and normal.

Answer 1

In order to make sense of the statement, we need an overarching field extension $F:K$ which contains $L$ and $M$ (otherwise, if $L$ and $M$ are just abstract field extensions, then what does their intersection actually mean?).

Finiteness is straightforward: simply observe that $L\cap M\subseteq L$ is a subspace of a finite dimensional vector space over $K$, and so must be finite dimensional.

For normality: if a $K$-irreducible polynomial $f$ does not split into linear factors in both $L$ and $M$, then by the normality of each it must have no roots in one of $L$ or $M$, and thus has no roots in $L\cap M$. Otherwise, $$f=(x-l_1)\cdots(x-l_d) = (x-m_1)\cdots (x-m_d).$$ Since each of these are factorizations of $f$ in $F[x]$, which is a UFD, the $l_i$ are just a reordering of the $m_j$, and thus $f$ splits into linear factors in $L\cap M$.