a is separable on F, b is separable on F(a), proof that b is separable on F

Question

$a$ is separable on $F$, $b$ is separable on $F(a)$, why $b$ is separable on $F$ ?

I tried using definitions：Let $\min(b,F(a))=f(x),$ $\min(b,F)=g(x)$.Then $g(x)|f(x)$,then I was stucked here.

Certainly $g(x)$ is separable on $F$,but $g(x)|f(x)$, I can't say that $f(x)$ is separable on $F$. And I don't know how to use $a$ is separable on $F$.