Use of Polynomials over Z/pZ in Number Theory

Wilson's theorem using factorization of $x^p - x$ over $\mathbb{F}_p$

Recall that $$x^p-x\equiv x(x-1)(x-2)\cdots(x-(p-1))\pmod{p}.$$ Thus, we have $$x^{p-1}-1\equiv (x-1)(x-2)\cdots(x-(p-1))\pmod{p}.$$

This can be derived from Fermat's Little Theorem ($\forall x\in\{1,2,\cdots,p-1\},x^{p-1}\equiv 1\pmod{p}$). The result follows after using the factor theorem $p-1$ times.

Let $x=p$. Then, we have $$p^{p-1}-1\equiv (p-1)(p-2)\cdots(p-(p-1))\pmod{p}.$$ $$0-1\equiv (p-1)(p-2)\cdots2\cdot1\pmod{p}.$$ $$-1=(p-1)!\pmod{p}$$ This result is called Wilson's theorem. Actually, it's only one direction. Wilson's theorem actually states that if this is true, then $p$ must be prime as well. Let's prove that direction.

We wish to prove that if $-1=(p-1)!\pmod{p}$, then $p$ is prime. Suppose for the sake of contradiction, that this is true, and $p$ is not prime. Then, we know $p=ab$ for some $1<a,b<p$ where $\gcd(a,b)=1$. However, this tells us that $a$ and $b$ are in $(p-1)!$. That means that $p=ab\mid(p-1)!$, so $(p-1)!\equiv 0\pmod{p}$, which is a contradiction. Thus, $p$ must be prime.