# Strict Prime-Intersective Polynomials for a Fixed Prime Number

5-2022

Thesis

## Degree Name

Master of Science (MS)

## College/School

College of Science and Mathematics

Mathematics

Aihua Li

Mark Korlie

Jonathan Cutler

## Abstract

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.

PDF

COinS