Let $F$ be a field and $\sigma:F[x]\to F[x]$ be automorphism, $\sigma(a) = a$ for all $a\in F$. I'm supposed to show that $\sigma(f(x)) = f(ax+b)$ for some $a\not = 0$ and $b$ in $F$.
Now I've got a solution that my professor gave me that seems to assume that the automorphism must have the form $\sigma(f(x)) = f(p(x))$ for some $p(x)\in F[x]$. So my question is how does $\sigma$ being an automorphism on $F[x]$ and $\sigma(a) = a$ for all $a\in F$ give us that $\sigma(f(x) = f(p(x))$ for some $p(x)\in F[x]$, why can't there be some weirder looking automorphism?
I've looked at Automorphisms of $F[x]$, however the only solution seems to make the same assumption that my professor makes.