Wilson's Theorem for Finite Fields

In this short note, we introduce an analogue of Wilson’s theorem for all nonzero elements $a_1,a_2,\mathrm{.}\mathrm{.}\mathrm{.},a_{q-1}$ of a finite filed $\mathbb{F}$ with $\mathrm{\mid }\mathbb{F}\mathrm{\mid }=q\ge 3$, as follows: $$\sum _{1\le i_1<i_2<\mathrm{.}\mathrm{.}\mathrm{.}<i_k\le q-1}{a}_{i_1}{a}_{i_2}\mathrm{.}\mathrm{.}\mathrm{.}{a}_{i_k}=\lfloor \frac{k}{q-1}\rfloor (-1{)}^q(k=1,2,\mathrm{.}\mathrm{.}\mathrm{.},q-1),$$ which the left hand side of above formula is the $k-$th elementary symmetric polynomial evaluated at $a_1,a_2,\mathrm{.}\mathrm{.}\mathrm{.},a_{q-1}$. Specially, letting $\mathbb{F}={\mathbb{Z}}_p$ with $p\ge 3$, reproves Wilson’s theorem and yields some Wilson type identities. Finally, we obtain an analogue of Wolstenholme’s theorem for nonzero elements of a finite filed.