Irreducibility criterion, irreducible factors, Newton polygon techniques

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 $(K,\nu)$.