What Are the Automorphisms of Z[x]?

[SS521]

Question: What are the automorphisms of Z[x]?

I know there are two automorphisms, one of which is the identity map, ø(f(x)) = f(x).

What is the other one? ø(f(x)) = -f(x) for all f(x) in Z[x]? Or does it have something to do with the degree or factorization of the polynomials? Please explain in detail because I don't know much about ring automorphisms. Thanks.

 

[economicsnerd]

Close: Try $\phi$ given by $\phi(f)(x)=f(-x)$.

More generally, if $R$ is any commutative ring, and $u\in R$ is any unit element, then $\phi_u$ given by $\phi_u(f)(x)=f(ux)$ is a ring automorphism.