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.