New Wilson-like theorems arising from Dickson polynomials

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+aare nonsquares} \right\} = 2 if $q\equiv \pm 1(\mathrm{m}\mathrm{o}\mathrm{d}12)$, or $-1$ otherwise.