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

TL;DR

This paper investigates the factorization degrees of binomials over characteristic zero fields and establishes a multiplicativity property of the n-Hartley condition for polynomials, linking their factorization structure to Hartley conditions.

Contribution

It proves that the degrees of factors of binomials are divisible by the least such degree and shows the multiplicative nature of the n-Hartley condition for coprime integers.

Findings

Degrees of factors of binomials are divisible by the minimal degree.

The n-Hartley condition is multiplicative for coprime integers.

Characterization of polynomial Hartley conditions based on factorization.

Abstract

We prove that, over a field of characteristic $0$, the degrees of factors of a binomial $t^n-\alpha$ 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.