# Homomorphisms of Polynomials Over Integral Domains

1. Jul 31, 2012

### jmjlt88

Let A be an integral domain.
If c ε A, let h: A[x] → A[x] be defined by h(a(x))=a(cx).

Prove that h is an automorphism iff c is invertible.

This one really had me stumped. I have a general idea of what the function is doing. Now, assuming that h is an automorphism, we want to show that there is some element u such that cu=uc=1. My idea was to use the fact that h is injective. That is, if a(cx)=b(cx), then a(x)=b(x). I believe that I can get this fact to imply that c is invertible. Right approach?

Any help would be great! Thanks! :)