Wilson's Theorem for Finite Fields

Abstract

In this short note, we introduce an analogue of Wilson&#39;s theorem for all nonzero elements $a_1,a_2,...,a_{q-1}$ of a finite filed $\mathbb{F}$ with $|\mathbb{F}|=q\geq 3$, as follows: $$ \sum_{1\leq i_1< i_2<...< i_k\leq q-1}a_{i_1}a_{i_2}... a_{i_k}=\left\lfloor\frac{k}{q-1}\right\rfloor(-1)^q\hspace{10mm}(k=1,2,..., q-1), $$ which the left hand side of above formula is the $k-$th elementary symmetric polynomial evaluated at $a_1,a_2,...,a_{q-1}$. Specially, letting $\mathbb{F}=\mathbb{Z}_p$ with $p\geq 3$, reproves Wilson&#39;s theorem and yields some Wilson type identities. Finally, we obtain an analogue of Wolstenholme&#39;s theorem for nonzero elements of a finite filed.