Date of Award
Master of Science (MS)
College of Science and Mathematics
Thesis Sponsor/Dissertation Chair/Project Chair
In this thesis, we examine intersective polynomials, which are polynomials with integer coefficients that have a root modulo any positive integer greater than 1. For any prime number p, a p-intersective polynomial is a polynomial with integer coefficients which has a root in Zp. We define a special type of p-intersective polynomial called strict p-intersective polynomial that can be factored as the product of a p-intersective polynomial and an irreducible polynomial mod p. The main results include methods of construction of strict p-intersective polynomials for certain prime numbers p and enumeration of such polynomials of certain degrees.
Chapter 1 gives the history and background of intersective polyomials. In Chapter 2, we explore irreducible polynomials over the field Zp, where p is prime. Those pintersective polynomials of degree ≤ 5 for p = 2, 3 and 5 are investigated. In chapter 3, we analyze strict p-intersective polynomials with focus on counting the number of polynomials over Zp that are strict p-intersective. The chapter ends with constructing and enumerating p-intersective polynomials with degrees 3,4, and 5. Lastly, we construct some special p-intersective and intersective polynomials in chapter 4.
Baello, Rob Rexler, "Strict Prime-Intersective Polynomials for a Fixed Prime Number" (2022). Theses, Dissertations and Culminating Projects. 1007.