$F(a,b)$ separable if $F(a)$ and $F(b)$ are separable

Question

Let $F$ be a field and $F(a)$, $F(b)$ simple algebraic separable extensions of $F$.

Is then $F(a,b)$ separable over $F$?

I was trying to use the transitivity of finite separable extensions (since $F(a)$ and $F(b)$ must be finite and so $F(a,b)$ too) but I am a bit stuck there since I don't know if $F(a,b)$ is separable over $F(a)$.

Answer 1

Note that $F(a,b) = F(a)(b)$. The polynomial $m(b,F(a))$ divides in particular $m(b,F)$. The latter is separable, so the former is too, and that shows that $F(a,b)/F(a)$ is separable.