Wilson's theorem for finite fields

In this short note, we introduce an analogue of Wilson's theorem for allnonzero elements $a_1,a_2,...,a_{q-1}$ of a finite filed $\mathbb{F}$ with$|\mathbb{F}|=q\geq 3$, as follows: $$ \sum_{1\leq i_1< i_2<...< i_k\leqq-1}a_{i_1}a_{i_2}...a_{i_k}=\left\lfloor\frac{k}{q-1}\right\rfloor(-1)^q\hspace{10mm}(k=1,2,...,q-1), $$ which the left hand side of above formula is the $k-$th elementarysymmetric polynomial evaluated at $a_1,a_2,...,a_{q-1}$. Specially, letting$\mathbb{F}=\mathbb{Z}_p$ with $p\geq 3$, reproves Wilson's theorem and yieldssome Wilson type identities. Finally, we obtain an analogue of Wolstenholme'stheorem for nonzero elements of a finite filed.