Hurwitz quaternion valued polynomials and their denominators

As Hamilton's quaternions are a generalization of the complex numbers, algebraists and number theorists had an early interest in defining a set of integral quaternions inside the Hamilton's quaternions, which is analogous to the Gaussian integers inside complex numbers. The preferable choice for integral quaternions are Hurwitz integral quaternions, as introduced by German mathematician Adolf Hurwitz, they are the set of all Lipschitz integers and half integers. This thesis is concerned with developing the properties and finding the basis of the Hurwitz integer valued polynomials.;In the following chapters we will properly define Hamilton's and Hurwitz quaternions and develop the background necessary in attempting to properly describe Hurwitz integer valued polynomials. Existing results will be examined and techniques necessary to extend these results to higher degrees of Hurwitz integer valued polynomials are developed.;The Hurwitz integers form a non-commutative algebra, and thus the familiar integer concepts such' as primes, division algorithm, and prime factorization do not directly generalize from Gaussian integers. They are carefully defined. Similarly a careful definition for polynomials over quaternions and a proper method of its evaluation are chosen.