Field automorphisms of extension generated by two coprime algebraic elements.

Question

Let $F$ be a field and let $a,b$ be algebraic over $F$ with $[F(a) : F] = n$ and $[F(b) : F] = m$ coprime. Let $\sigma \in \textrm{Aut}(F(a,b)/F)$. Is it true that $\sigma(F(a)) = F(a)$ and $\sigma(F(b)) = F(b)$?

I tried letting $\alpha \in F(a)$ and considering the minimal polynomial of $\sigma(\alpha)$ over $F$ and looking at degrees, but this didn't help.

Answer 1

Try $a=\sqrt[3]{2}$, $b=\frac{-1+i\sqrt{3}}{2}$.

Note that $a,ab,ab^2$ are the three roots of $X^3-2$. In particular, there is an automorphism sending $a$ to $ab$.