 Solution

Compare constant terms in the polynomial identity proved above. The constant term of $X^{p-1}-1$ is $-1$, while that of the product is $(-1{)}^{p-1}(p-1)!$. For odd $p$, this gives $(p-1)!\equiv -1(\mathrm{mod}p)$. For $p=2$, the same conclusion holds because $1\equiv -1(\mathrm{mod}2)$. Thus Wilson's theorem follows: