Let R be a commutative ring with identity.

Let be a polynomial in R[x].

p(x) is a unit if and only if unit and are nilpotent.

So I can easily prove the first term, is a unit.

However I feel that I cannot guarantee that is nilpotent.

Since p is a unit, then there exists a q in R[x] (let's say the coefficients of q are of the form and that it has degree m) where p*q = 1.

So taking the product of the 2 polynomials:

and it must equal zero because a_0 is a unit, which implies that the last term = 1, and the rest equals 0.

So I can show that a_n is a zero divisor but I don't feel I can guarantee that it's nilpotent.

The hint says to look at the x^(n-1) term and multiply by a_n.

then the term on the right dissapears because a_nb_m =0.