论文标题
不可约性标准，不可减至因素，牛顿多边形技术
Irreducibility criterion, irreducible factors, Newton polygon techniques
论文作者
论文摘要
jakhar表明，对于$ f（x）= a_nx^n+ a_ {n-1} x^{n-1}+ \ cdot+ a_0 $（$ a_0 \ neq 0 $）是具有理性系数的多项式 (n-i)ν_p(a_0)> 0$ for every $0\le i\le n-1$, then $f(x)$ has at most $gcd(ν_p(a_0),n)$ irreducible factors over the field $\mathbb{Q}$ of rational numbers and each irreducible factor has degree at least $n/gcd(ν_p(a_0),n)$.本文的目的是在以下上下文中概括此标准：让$（k，ν）$为一个排名一个离散的有价值字段，$r_ν$其估值环和$ \ Mathbb {f}_ν$其残基字段。假设$ f（x）= ϕ^n（x）+ a_ {n- 1}（x）ϕ^{n-1}（x）+ \ cdot+ cdot+ a_0（x）\ inr_ν[x] $，每个$ i = 0，n-dots，n-1 $，n-1 $，n-1 $，$ a_i（x）在$ \ mathbb {f}_ν[x] $中，r_ν[x] $ inr_ν[x] $中的多项式$ ϕ \不可记录。如果每$ 0 \ le i \ le n-1 $，$nν_p（a_i）\ ge（n-i）ν_p（a_0）> 0 $，}然后，$ f（x）$最多具有$ gcd（ν_p（a_0（x）），n），$ gcd（a_0（x），n）$ yourdreducible因素，则是$ k^h $ and $ k^h $ cubifuc trecribuce trecribuc trect trecred trect trect trect trect trect trect in Irred Irred Irred Irred Irred Irred Irred Irred因素$ n/gcd（ν_p（a_0），n）$，其中$ k^h $是$（k，ν）$的henselization。
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 $ν_p(a_n)=0$ and $nν_p(a_i)\ge (n-i)ν_p(a_0)> 0$ for every $0\le i\le n-1$, then $f(x)$ has at most $gcd(ν_p(a_0),n)$ irreducible factors over the field $\mathbb{Q}$ of rational numbers and each irreducible factor has degree at least $n/gcd(ν_p(a_0),n)$. The goal of this paper is to generalize this criterion in the following context: Let $(K,ν)$ be a rank one discrete valued field, $R_ν$ its valuation ring and $\mathbb{F}_ν$ its residue field. Assume that $f(x)=ϕ^n(x) + a_{n- 1}(x)ϕ^{n-1}(x)+\cdot+ a_0(x)\in R_ν[x]$, with for every $i=0,\dots,n-1$, $a_i(x)\in R_ν[x]$, and $a_0(x)\neq 0$ for some monic polynomial $ϕ\in R_ν[x]$ with $\overlineϕ$ is irreducible in $\mathbb{F}_ν[x]$. If for every $0\le i\le n-1$, $nν_p(a_i)\ge (n-i)ν_p(a_0)>0$,} then $f(x)$ has at most $gcd(ν_p(a_0(x)),n)$ irreducible factors over the field $K^h$ and so over $K$ and each irreducible factor has degree at least $n/gcd(ν_p(a_0),n)$, where $K^h$ is the henselization of $(K,ν)$.
