What does F(a)(b) mean in the context of extension fields?

Question

I understand that F(a) is the smallest subfield that contains both F and a. What is the definition of F(a)(b)? I'm supposed to prove that it's equal to F(a,b), but without knowing how F(a)(b) is defined, I'm at a loss where to begin.

Answer 1

Let $K=F(a)$. Then $F(a)(b)=K(b)$. A slightly clearer notation might be $(F(a))(b)$ but some authors despise parenthesis overload.