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Strict Prime-Intersective Polynomials for a Fixed Prime Number

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Title: Strict Prime-Intersective Polynomials for a Fixed Prime Number
Authors: Baello, Rob Rexler
Summary: 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.
URL: https://digitalcommons.montclair.edu/etd/1007
Database: OpenDissertations
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An: ddu.oai.digitalcommons.montclair.edu.etd.2009
AccessLevel: 6
PubType: Dissertation/ Thesis
PubTypeId: dissertation
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  Label: Title
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  Data: Strict Prime-Intersective Polynomials for a Fixed Prime Number
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  Label: Authors
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  Data: <searchLink fieldCode="AR" term="%22Baello%2C+Rob+Rexler%22">Baello, Rob Rexler</searchLink>
– Name: Abstract
  Label: Summary
  Group: Ab
  Data: 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.
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  Data: <link linkTarget="URL" linkTerm="https://digitalcommons.montclair.edu/etd/1007" linkWindow="_blank">https://digitalcommons.montclair.edu/etd/1007</link>
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RecordInfo BibRecord:
  BibEntity:
    Languages:
      – Code: eng
        Text: English
    Subjects:
      – SubjectFull: Intersective polynomial
        Type: general
      – SubjectFull: p-intersective polynomial
        Type: general
      – SubjectFull: Strict p-intersective polynomial
        Type: general
      – SubjectFull: Irreducible polynomial
        Type: general
      – SubjectFull: Mathematics
        Type: general
      – SubjectFull: Polynomials, Numbers, Prime, Irreducible polynomials
        Type: general
    Titles:
      – TitleFull: Strict Prime-Intersective Polynomials for a Fixed Prime Number
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Baello, Rob Rexler
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 05
              Type: published
              Y: 2022
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