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The degrees of irreducible factors of binomials and multiplicativity of the n-Hartley condition

Matthew Bolan, Ben Williams

TL;DR

This work addresses the problem of understanding the degrees of irreducible factors of the binomial $t^n-\alpha$ over a field of characteristic $0$ and its connection to the $n$-Hartley condition for integer polynomials. The authors define $\lambda_K(\alpha,n)$ as the minimal degree of an irreducible factor of $t^n-\alpha$, and prove that every factor degree is a multiple of $\lambda_K(\alpha,n)$; they further establish multiplicativity: $\lambda_K(\alpha,m)\lambda_K(\alpha,n)=\lambda_K(\alpha,mn)$ for coprime $m,n$. These results imply that, for primitive $\Delta(t)=h(t)^s$, $\Delta$ is $n$-Hartley iff $\lambda(\alpha,n)\mid s$, and that $\Delta$ is $mn$-Hartley iff it is $m$-Hartley and $n$-Hartley when $\gcd(m,n)=1$. The work links binomial factorization properties, Galois-theoretic orbit analysis, and the Hartley condition to implications for the Alexander polynomial of freely periodic knots, enriching the connection between algebraic factorization and knot symmetry.

Abstract

We prove that, over a field of characteristic $0$, the degrees of factors of a binomial $t^n-α$ are divisible by the least such degree. As a consequence, we deduce that for relatively prime natural numbers $m,n$, a polynomial has the $mn$-Hartley condition if and only if it has the $m$-Hartley and $n$-Hartley conditions.

The degrees of irreducible factors of binomials and multiplicativity of the n-Hartley condition

TL;DR

This work addresses the problem of understanding the degrees of irreducible factors of the binomial over a field of characteristic and its connection to the -Hartley condition for integer polynomials. The authors define as the minimal degree of an irreducible factor of , and prove that every factor degree is a multiple of ; they further establish multiplicativity: for coprime . These results imply that, for primitive , is -Hartley iff , and that is -Hartley iff it is -Hartley and -Hartley when . The work links binomial factorization properties, Galois-theoretic orbit analysis, and the Hartley condition to implications for the Alexander polynomial of freely periodic knots, enriching the connection between algebraic factorization and knot symmetry.

Abstract

We prove that, over a field of characteristic , the degrees of factors of a binomial are divisible by the least such degree. As a consequence, we deduce that for relatively prime natural numbers , a polynomial has the -Hartley condition if and only if it has the -Hartley and -Hartley conditions.
Paper Structure (2 sections, 8 theorems, 22 equations)

This paper contains 2 sections, 8 theorems, 22 equations.

Table of Contents

  1. The n-Hartley condition
  2. Proof of Theorem \ref{['th:main']}

Key Result

Proposition 1.1

If $h(t) \in \mathbb{Z}[t]$ is an irreducible primitive polynomial of positive degree having $\alpha$ as a root and $s$ is a nonnegative integer, then $\Delta(t)=h(t)^s$ has the $n$-Hartley property if and only if there exists some polynomial $q(t) \in \mathbb{Q}(\alpha)[t]$ of degree $s$ all of who

Theorems & Definitions (16)

  • Proposition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • Theorem 1.6
  • proof
  • Definition 2.1
  • ...and 6 more