TITLE: Let $L,F$ be extensions over the field $K$ and $L,F$ are contained in a common field (cont)
QUESTION [5 upvotes]: Let $L,F$ be extensions over the field $K$ and $L,F$ are contained in a common field. Prove that $\left[LF:K\right]\leq\left[L:K\right]\left[F:K\right]$
Help me.
Thanks a lot.

REPLY [3 votes]: Assume without loss of generality that $[L:K] = n < \infty$, and hence
$$
L = K[\alpha_1, \alpha_2, \ldots , \alpha_n] \Rightarrow LF = F[\alpha_1, \alpha_2, \ldots , \alpha_n]
$$
and so $[LF:F] \leq [L:K]$. Thus,
$$
[LF:K] = [LF:F][F:K] \leq [L:K][F:K]
$$