Problem: Let $R$ be a domain. Prove that if a polynomial in $R[x]$ is a unit, then it is a nonzero constant (the converse is true if $R$ is a field).
My attempt: We proof that by contradiction. Suppose $u \in R$ be a unit of $R$ but $u$ is not a nonzero constant, then we have $\deg (u) > 0$.
$\forall f \in R$, $\deg (uf) \leq \deg (u) + \deg (f)$.
On the other hand, $u$ is a unit of $R$, so $\deg (uf) = \deg (f)$.
Associate with the inequality above, we see that the equality hold if and only if $\deg (u) = 0$. So we can conclude that $u$ is a nonzero constant.
Is my proof correct??? Thanks all!!!
Well, a proof by contradition is not necessary. Suppose $f$ is a unit with inverse $g$. Then $fg=1$. Using degrees, we obtain $$0 = {\rm deg}(1) = {\rm deg}(fg) = {\rm deg}(f) + {\rm deg}(g).$$ The last equality holds since $R$ has no zero divisors. As the degree is a nonnegative function, it follows that both, $f$ and $g$ have degree zero and so are constants (elements of $R$).