arXiv 2007.076592020-07-15math.NT

不可约准则，不可约因子，牛顿多边形技术

Irreducibility criterion, irreducible factors, Newton polygon techniques

Lhoussain El Fadil

摘要：

Jakhar证明了，对于$f(x)=a_nx^n + a_{n-1}x^{n-1}+\cdot+ a_0$（$a_0\neq 0$）是一个有理系数多项式，如果存在一个素数$p$满足$ν_p(a_n)=0$和$nν_p(a_i)\ge (n-i)ν_p(a_0)> 0$，则$f(x)$在有理数域$\mathbb{Q}$上至多有$gcd(ν_p(a_0),n)$个不可约因子，且每个不可约因子的次数至少为$n/gcd(ν_p(a_0),n)$ .本文的目的是推广这一准则在以下情况下：令$(K,ν)$是一个秩为1的离散值域，$R_ν$的赋值环和$\mathbb{F}_ν$的剩余域。假设$f(x)=ϕ^n(x) + a_{n- 1}(x)ϕ^{n-1}(x)+\cdot+ a_0(x)\in R_ν[x]$，对于每个$i=0,\dots,n-1$，$a_i(x)\in R_ν[x]$和$a_0(x)\neq 0$，对于某个一元多项式$ϕ\in R_ν[x]$和$\overlineϕ$，在$\mathbb{F}_ν[x]$中不可约。如果对每个$0\le i\le n-1$，$nν_p(a_i)\ge (n-i)ν_p(a_0)>0$，}，则$f(x)$在域$K^h$上至多有$gcd(ν_p(a_0(x)),n)$不可约因子，因此在$K$上，每个不可约因子的度至少为$n/gcd(ν_p(a_0),n)$，其中$K^h$是$(K,ν)$的Henselization。

英文摘要：

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 $(K,ν)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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