If $F(a) \subseteq F(c)$, then $F(c)$ is an extension of $F(a)$

Question

How do I prove the statement above? I have noticed my textbook using it but I cannot find an explanation of why that is true. I can see why this is true in things like $Q(\sqrt[4]{2})$ and $Q(\sqrt{2})$ but that is because I can find the polynomial that makes it an extension. I am pretty sure I have to assume $a,c$ are algebraic but i am not sure. Any help would be appreciated.

Answer 1

Yes, because $F(a)\subseteq F(c)$ implies $F(a)(c)=F(c)$. After all, $F(c)\subseteq F(c)(a)=F(a,c)=F(a)(c)$. On the other hand, since $a\in F(c)$ since $F(a)\subseteq F(c)$, so $F(a)(c)=F(a,c)\subseteq F(c)$.