Explain proof of irreducibility of $x^{p-1} + 2x^{p-2}+ \dots +(p-1)x + p$
Solution 1:
Let the complex zeroes of $Q(x)$ be $r_1,\ldots,r_m$, not necessarily distinct. Then $Q(x)=\prod_{k=1}^m(x-r_k)$, and $1=|Q(0)|=\left|(-1)^k\prod_{k=1}^mr_k\right|$. But $\left|(-1)^k\prod_{k=1}^mr_k\right|=\prod_{k=1}^m|r_k|>1$, so this is impossible.