Intersective polynomials and Hensel's Lemma

by Andrea Hyde

Institution: University of British Columbia
Department: Mathematics
Degree: MS- MSc
Year: 2014
Record ID: 2029705
Full text PDF: http://hdl.handle.net/2429/46574


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 odd prime q. In the last chapter, we treat some special cases of intersective polynomials which will be considered in detail in future work.