A Generalization of Wilson's Theorem

Abstract

Wilson’s theorem states that if p is a prime number then (p−1)! ≡ −1 (mod p). One way of proving Wilson’s theorem is to note that 1 and p − 1 are the only self-invertible elements in the product (p − 1)!. The other invertible elements are paired off with their inverses leaving only the factors 1 and p−1. Wilson’s theorem is a special case of a more general result that applies to any finite abelian group G. In order to apply this general result to a finite abelian group G, we are required to know the self-invertible elements of G. In this thesis, we consider several groups formed from polynomials in quotient rings. Knowing the self-invertible elements allows us to state Wilson-like results for these groups. Knowing the order of these groups allows us to state Fermat-like results for these groups. The required number theoretical background for these results is also included.