What are the automorphisms $\sigma$ of $F[x]$ with the property that $ \sigma(f) =f $ for every $f \in F$?
$F$ is a field here.
What are the automorphisms $\sigma$ of $F[x]$ with the property that $ \sigma(f) =f $ for every $f \in F$?
$F$ is a field here.
I'm assuming that $F$ is a field. An $F$-homomorphism $\sigma\colon F[x] \to F[x]$ is determined by the polynomial $\sigma(x)$, and the image will be $F[\sigma(x)] \subset F[x]$. Thus $\sigma$ is surjective if and only if we can write $x$ as a polynomial in $\sigma(x)$. Can this be done if $\sigma(x)$ has degree $\geq 2$? Is it possible to write down an inverse automorphism if $\sigma(x) = ax + b$ for some $a \in F^*$ and $b \in F$?