I thought I would try to answer a question raised by your comment. A version of the following lemma is true in any field, but I will formulate it for the integers modulo $p$.
Lemma Let $P(x)$ and $Q(x)$ be monic (lead coefficient $1$) polynomials of degree $d$. If $P(a) \equiv Q(a) \pmod{p}$ for $d$ incongruent numbers $a$, then the corresponding coefficients of the two polynomials are congruent modulo $p$.
Now look at the two polynomials $P(x)=(x-1)(x-2)\cdots(x-p+1)$ and $Q(x)=x^{p-1}-1$. They are both monic of degree $p-1$. Clearly $P(a) \equiv 0$ for $a=1,2,\dots,p-1$. Also, $Q(a)\equiv 0$ at the same places, by Fermat's Theorem.
So corresponding coefficients are congruent to each other. Note in particular that if $p$ is an odd prime, then the constant term of $P$ is $(p-1)!$. The constant term of $Q$ is $-1$, and we conclude that $(p-1)! \equiv -1 \pmod{p}$, Wilson's Theorem! Take any other coefficient of $P$ except the coefficient of $x^{p-1}$. The corresponding coefficient in $Q$ is $0$, which tells us that all the coefficients of $P$ are congruent to $0$ except for the first and the last.