  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.
