MSc Math Defence: A Generalization of Wilson's Theorem by Thomas Jeffery

CANDIDATE:   THOMAS JEFFERY

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. Here when we say an element is invertible we mean invertible modulo the prime p. 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.

Fermat’s little theorem states that if p is prime, a is an integer, and p Xa then ap−1  ≡ 1 (mod p).  Fermat’s little theorem is a special case of a more general result that applies to any finite abelian group G.  This general result can be proved by noting that when you multiply each element in a finite abelian group by a group element you just permute the group. In order to apply this to a finite abelian group G, we are required to know the order 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. This background includes quadratic residues, primes in residue classes, binary quadratic forms, primitive congruence roots, and sums of squares.