Intersective polynomials and Hensel's Lemma
department or school: Arts and Sciences, Irving K. Barber School of (Okanagan)
department or school: Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
data provider: DSpace
degree campus: UBCO
creator: Hyde, Andrea
issued: 2014-04-24T17:46:30Z
issued: 2014
related degree: Master of Science - MSc
degree grantor: University of British Columbia
description: An intersective polynomial is a polynomial with integer coefficients that has no rational roots, but has a root modulo every integer greater than 1. These polynomials have been difficult to find using traditional methods. In this thesis, we employ elementary methods, namely Hensel’s Lemma and the Chinese remainder theorem, to allow us to create three new infinite families of intersective polynomials. In order to create a candidate intersective polynomial, we employ methods from Galois theory. We multiply together carefully chosen polynomials that define subfields of a splitting field to create our candidate. We chose the subfields by first finding the n-cover, a collection of n proper subgroups, of the Galois group and identifying the corresponding subfields. We multiply the minimal polynomials of the subfields of the chosen splitting fields together to create the candidate intersective polynomial. From our method of creating candidates we can see that intersective polynomials have at least two irreducible factors. In order to prove that these polynomials have a root modulo every integer greater than 1, we examine each factor modulo certain prime numbers and determine which sets of primes admit solutions for each factor. We then use Hensel’s Lemma to lift those solutions to infinitely high powers of the particular primes. Finally, we combine the solutions modulo prime powers by using the Chinese remainder theorem to show that the polynomial has solutions modulo every integer greater than 1. Through the method outlined above, we have created three infinite families of intersective polynomials: (x³ − n)(x² + 3) when the prime factors of n are of the form 3k + 1 and n is congruent to 1 mod 9; (x² − a)(x² − b)(x² − a₁b₁) when at least one of a, b, a₁b₁ is congruent to 1 modulo 8 and one of the Legendre Symbols (a/p), (b/p), (a₁b₁/p) has the value +1; and (x^q −n)(x^(q-¹) +x^q-²+...+ x + 1) when the prime factors of n are kq + 1 and n is congruent to 1 modulo q² for an o…
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/46574
note: Intersective Polynomials and Hensel’sLemmabyAndrea May HydeB.Sc. Hons., The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)April 23 2014c© Andrea May Hyde, 2014AbstractAn intersective polynomial is a polynomial with integer coefficients that has no rationalroots, but has a root modulo every integer greater than 1. These polynomials have beendifficult to find using traditional methods. In this thesis, we employ elementary methods,namely Hensel’s Lemma and the Chinese remainder theorem, to allow us to create threenew infinite families of intersective polynomials.In order to create a candidate intersective polynomial, we employ methods from Ga-lois theory. We multiply together carefully chosen polynomials that define subfields of asplitting field to create our candidate. We chose the subfields by first finding the n-cover,a collection of n proper subgroups, of the Galois group and identifying the correspondingsubfields. We multiply the minimal polynomials of the subfields of the chosen splittingfields together to create the candidate intersective polynomial.From our method of creating candidates we can see that intersective polynomials haveat least two irreducible factors. In order to prove that these polynomials have a root moduloevery integer greater than 1, we examine each factor modulo certain prime numbers anddetermine which sets of primes admit solutions for each factor. We then use Hensel’sLemma to lift those solutions to infinitely high powers of the particular primes. Finally,we combine the solutions modulo prime powers by using the Chinese remainder theoremto show that the polynomial has solutions modulo every integer greater than 1.Through the method outlined above, we have created three infinite families of intersec-tive polynomials: (x3 − n)(x2 + 3) when the prime factors of n are of t…
has type: Thesis/Dissertation
date issued: 2014-05
is shown at: 10.14288/1.0074336
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivs 2.5 Canada
rights URI: http://creativecommons.org/licenses/by-nc-nd/2.5/ca/
scholar level: Graduate
type: Text
identifier URI: http://hdl.handle.net/2429/46574