Irreducibility criterion, irreducible factors, Newton polygon techniques

TL;DR

This paper generalizes a polynomial irreducibility criterion from rational coefficients to polynomials over rank one discrete valued fields, using Newton polygon techniques to bound the number and degree of irreducible factors.

Contribution

It extends Jakhar's irreducibility criterion to polynomials over valuation rings of rank one discrete valued fields, incorporating Newton polygon methods.

Findings

Provides bounds on the number of irreducible factors over henselization and base field.

Establishes degree lower bounds for irreducible factors.

Generalizes existing criteria from rational coefficients to valuation rings.

Abstract

Jakhar shown that for $f(x)=a_nx^n + a_{n-1}x^{n-1}+\cdot+ a_0$ ($a_0\neq 0$) is a polynomial with rational coefficients, if there exists a prime integer $p$ satisfying $\nu_p(a_n)=0$ and $n\nu_p(a_i)\ge (n-i)\nu_p(a_0)> 0$ for every $0\le i\le n-1$, then $f(x)$ has at most $gcd(\nu_p(a_0),n)$ irreducible factors over the field $\mathbb{Q}$ of rational numbers and each irreducible factor has degree at least $n/gcd(\nu_p(a_0),n)$. The goal of this paper is to generalize this criterion in the following context: Let $(K,\nu)$ be a rank one discrete valued field, $R_\nu$ its valuation ring and $\mathbb{F}_\nu$ its residue field. Assume that $f(x)=\phi^n(x) + a_{n- 1}(x)\phi^{n-1}(x)+\cdot+ a_0(x)\in R_\nu[x]$, with for every $i=0,\dots,n-1$, $a_i(x)\in R_\nu[x]$, and $a_0(x)\neq 0$ for some monic polynomial $\phi\in R_\nu[x]$ with $\overline{\phi}$ is irreducible in $\mathbb{F}_\nu[x]$. If for every $0\le i\le n-1$, $n\nu_p(a_i)\ge (n-i)\nu_p(a_0)>0$,} then $f(x)$ has at most $gcd(\nu_p(a_0(x)),n)$ irreducible factors over the field $K^h$ and so over $K$ and each irreducible factor has degree at least $n/gcd(\nu_p(a_0),n)$, where $K^h$ is the henselization of $(K,\nu)$.