Homomorphisms of Polynomials Over Integral Domains

[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! :)

 

[lippyka]

Yes, your approach is correct. Now, for the converse, assume c is invertible and let u be its inverse. Let a(x), b(x) ∈ A[x]. Then we have h(a(x))=a(cx) and h(b(x))=b(cx). If h(a(x))=h(b(x)), then a(cx)=b(cx). Applying u to both sides, we get a(x)=b(x). Thus, h is injective. To show that h is surjective, let y(x) ∈ A[x]. Then y(x) = y(u·cx). Let a(x) = u·y(x). Then h(a(x)) = a(cx) = u·y(cx) = y(x). Thus, h is surjective. Since h is both injective and surjective, it is an automorphism.