 Solution

Lagrange theorem for polynomial congruences says that a nonzero polynomial of degree $d$ over a field has at most $d$ distinct roots of a polynomial. It follows by repeatedly factoring out $x-a$ at a root. Over the finite field ${\mathbb{F}}_p$, the only solutions of $x^2=1$ are $1$ and $-1$: these are distinct because $p$ is odd, and the root bound excludes any others.

Every nonzero element has a unique multiplicative inverse. Pair each element of ${\mathbb{F}}_p^{\times }$ with its inverse. All pairs of distinct elements contribute $1$ to the product; the only unpaired elements are $1$ and $-1$. Thus Wilson theorem follows:If Lagrange's theorem is formulated for finite groups, it states that the order of a subgroup divides the order of the group; in particular every element of ${\mathbb{F}}_p^{\times }$ has order dividing $p-1$. The inverse-pairing proof above establishes the required congruence directly.