Wilson's theorem

TL;DR

This paper generalizes Wilson's theorem to the multiplicative groups of quotient rings of integers in number fields, providing a complete classification of possible product values with simpler proofs.

Contribution

It offers a new, more straightforward proof of a generalization of Wilson's theorem for algebraic number theory contexts.

Findings

Identifies four possible product values for the group elements

Provides necessary and sufficient conditions for each case

Simplifies the proof of the generalization

Abstract

We show that there are four possibilities for the product of all elements in the multiplicative group of a quotient of the ring of integers in a number field, and give precise conditions for each of the possibilities to occur. This generalisation of Wilson's theorem turns out to have been first discovered by M. La\v{s}\v{s}\'ak (2000), but our proof is simpler and more direct.