Irreducibility criterion, irreducible factors, Newton polygon techniques

Abstract

Jakhar shown that for $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdot +a_0$ ($a_0\ne 0$) 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 $f(x)={\varphi }^n(x)+a_{n-1}(x){\varphi }^{n-1}(x)+\cdot +a_0(x)\in R_{\nu }[x]$, with for every $i=0,\dots ,n-1$, $a_i(x)\in R_{\nu }[x]$, and $a_0(x)\ne 0$ for some monic polynomial $\varphi \in R_{\nu }[x]$ with $\overline{\varphi }$ 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 $(K,\nu )$.