The condition that $[LM:K]=[L:K][M:K]$ holds in the field.

Question

Let $L$ and $M$ be intermediate fields of the extension $K\subset F$, of finite dimension over $K$. Assume that $L\cap M=K$ and $[L:K]$ or $[M:K]$ is 2, then $[LM:K]=[L:K][M:K]$ holds.

How can I use the condition that $[L:K]$ or $[M:K]$ is 2? Any help, please.

Answer 1

1) Show that $[LM : K] \le [L : K][M : K]$.

2) WLOG let $[L : K] = 2$. Show that $[LM : K]/[M : K]$ is either $1$ or $2$ (note: $M \subseteq LM$).

3) If $[LM : K] = [M : K]$, show that $L \subseteq M$. Deduce that $K = L \cap M = L$, contradicting $[L : K] = 2$.