divisibility of polynomials over a finite field

Question

Let $K$ be a finite field.

Let $f$ be an irreducible polynomial over $K$ of degree $n\ge 2$.

Let $P_i$ be the product of all polynomials of degree $i$ over $K$.

Prove that $f$ divides $P_1\cdot\ldots\cdot P_{n-1}-1$.

Answer 1

This is essentially Wilson's theorem for $ K[X]/(f) $. To prove it, simply note that any polynomial of degree $ 1 \leq \deg < n $ will admit an inverse of degree $ 1 \leq \deg < n $ modulo $ f $, and no such polynomial is self-inverse, so you may pair together inverses in the product to obtain that every factor in the product cancels, leaving you with $ 1 $ modulo $ f $.