On the factorization of $x^{d_{r}} + x^{d_{r-1}} + \cdots + x^{d_{2}} + x^{d_{1}} + p$ and a conjecture of C. Nicol

For an arbitrary prime $p$ and arbitrary positive integers $r$ and $d_{1}, \ldots, d_{r}$ with $d_{r} > \cdots > d_{1}$, we show that $x^{d_{r}} + x^{d_{r-1}} + \cdots + x^{d_{2}} + x^{d_{1}} + p$ removed of all of its monic irreducible reciprocal factors is irreducible. As a consequence, we establi