2020/07/15 by Lhoussain El Fadil, Fadil, Lhoussain El
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2007.07659
Jakhar shown that for f(x)=anxn + an-1xn-1+⋅+ a0 (a0≠ 0) is a polynomial with rational coefficients, if there exists a prime integer p satisfying νp(an)=0 and nνp(ai)≥ (n-i)νp(a0)> 0 for every 0≤ i≤ n-1, then f(x) has at most gcd(νp(a0),n) irreducible factors over the field ℚ of rational numbers and each irreducible factor has degree at least n/gcd(νp(a0),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, Rν its valuation ring and \mathbbFν its residue field. Assume that f(x)=ϕn(x) + an- 1(x)ϕn-1(x)+⋅+ a0(x)∈ Rν[x], with for every i=0,…,n-1, ai(x)∈ Rν[x], and a0(x)≠ 0 for some monic polynomial ϕ∈ Rν[x] with ϕ is irreducible in \mathbbFν[x]. If for every 0≤ i≤ n-1, nνp(ai)≥ (n-i)νp(a0)>0, then f(x) has at most gcd(νp(a0(x)),n) irreducible factors over the field Kh and so over K and each irreducible factor has degree at least n/gcd(νp(a0),n), where Kh is the henselization of (K,ν).