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Some factorization results on polynomials having integer coefficients

Jitender Singh, Rishu Garg

Abstract

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing some sufficiency conditions on their coefficients along with some conditions on the prime factorization of constant term or the leading coefficient of the underlying polynomials in conjunction with the information about their root location.

Some factorization results on polynomials having integer coefficients

Abstract

In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing some sufficiency conditions on their coefficients along with some conditions on the prime factorization of constant term or the leading coefficient of the underlying polynomials in conjunction with the information about their root location.
Paper Structure (3 sections, 11 theorems, 24 equations)

This paper contains 3 sections, 11 theorems, 24 equations.

Table of Contents

  1. Introduction.
  2. Main results
  3. Proofs of Theorems

Key Result

Theorem 1

Let $f=a_0+ a_{1}z+\cdots+a_n z^m\in \Bbb{Z}[z]$ be a polynomial and suppose there is a prime $p$ such that $p$ does not divide $a_m$, $p$ divides $a_i$ for $i=0,\ldots,m-1$, and for some $k$ with $0\leq k\leq m-1$, $p^{2}$ does not divide $a_k$. Let $k_0$ be the smallest such value of $k$. If $f(z)

Theorems & Definitions (21)

  • Theorem 1: Weintraub W
  • Theorem 2
  • proof
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Corollary 7
  • ...and 11 more