Challenge question:

Let $\displaystyle R$ be a commutative ring and $\displaystyle f(x)=a_0+...+a_nx^n \in R[x]$ then $\displaystyle f$ is a unit if and only if $\displaystyle a_0$ is a unit and $\displaystyle a_1,...a_n$ are nilpotent.

I have a solution using more involved ring theory, but I'm really interested to see more elementary solutions (or even more involved or indirect, why not).

Moderator approved CB