New Wilson-like theorems arising from Dickson polynomials

TL;DR

This paper introduces new Wilson-like theorems by deriving formulas for products of specific subsets of finite field elements, expanding classical results with simple, natural descriptions of these sets.

Contribution

It presents novel formulas for products over particular subsets of finite fields, generalizing Wilson's theorem with natural set definitions and simple product expressions.

Findings

Formulas for products over subsets where 1-a and 3+a are nonsquares

Explicit product values depending on congruences modulo 12

Extension of Wilson's theorem to new classes of subsets

Abstract

Wilson's Theorem states that the product of all nonzero elements of a finite field ${\mathbb F}_q$ is $-1$. In this article, we define some natural subsets $S \subset {\mathbb F}_q^\times$ and find formulas for the product of the elements of $S$, denoted $\prod S$. These new formulas are appealing for the simple, natural description of the sets $S$, and for the simplicity of the product. An example is $\prod\left\{ a \in {\mathbb F}_q^\times : \text{$1-a$ and $3+a$ are nonsquares} \right\} = 2$ if $q \equiv \pm 1 \pmod{12}$, or $-1$ otherwise.