Fixed Divisors and Generalized Factorials

Abstract

This thesis contains four chapters and is devoted to the study of the xed divison a polynomial. In th first chapter we recall a few of the known/unknown results in Algebra. In the second chapter, we compile concisely the work done by various mathematicians on computing and bounding divisors in various algebraic settings.At the end of this chapter, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of nn matrices over Z (or Dedekind domain) could lead to the generalization of xed divisors in that setting. In the third chapter we present the applications of xed divisors to various algebraic and number theoretic problems.The fourth chapter is an extension of our work on xed divisors. The work is presented in an orderly fashion so as to start from the simplest case of Z; progressively leading up to the case of Dedekind domains. We also ask a few open questions according to theircontext, which may give impetus to the reader to work further in this direction.