I've a question: How do I proof that if A is an integral domain, then the units of A[x] are the constant polynomials that are units of A? I'm really confused here and I'd be really glad if someone could show me how to prove it by explaining the steps...
Many thanks in advance!!!
The first part is to prove that the units of $A$ are units of $A[x]$, this part is easy because the inverse of the constant polynomial $a$ is the constant polynomial $a^{-1}$.
Now, suppose $P$ is a unit and suppose $Q$ is its inverse. Since $A$ is an integral domain there are no zero divisors and so we can easily prove the identity $\deg(R_1R_2) = \deg(R_1) + \deg(R_2)$ for arbitrary polynomials $R_1$ and $R_2$.
It follows $0 = \deg(PQ) = \deg(P) + \deg(Q)$. We conclude $P$ and $Q$ must be constant polynomials, suppose they are the constant $p$ and $q$ polynomials. It follows $pq=1$ and so $p$ is a unit of $A$.