A resultant condition for the irreducibility of the polynomials
type: Item
instance of: scholarly article
title: A resultant condition for the irreducibility of the polynomials
zbMATH Open document ID: 0615.13002
DOI: 10.1016/0022-314X(87)90018-7
author: Laurenţiu Panaitopol
author: Doru Ştefănescu
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Main result (theorem 1): Let (K,v) be a discrete valued field, A the valuation ring of v, \(\bar K\) its residue field, p in K such that \(v(p)=1\) and X an indeterminate over K. Let F be a monic polynomial in A[X] such that \(F=f_ 1^{m_ 1}...f_ n^{m_ n}+pg\) where \(f_ 1,...,f_ n\) and g are in \(A[X]\), \(f_ 1,...,f_ n\) are monic, \(\bar f{}_ 1,...,\bar f_ n\) are nonconstant and irreducible in \(\bar K[X]\) and \(\bar f_ i\) does not divide \(\bar g\) if \(m_ i\) is not less then 2. Let \(h=f_ 1^{m_ 1}...f_ n^{m_ n}.\) If, for every nontrivial factorization \(h=h_ 1h_ 2\), there is \(j=1,2\) and \(i=1,...,n\) such that \(f_ i\) divides \(h_ j\) and for all divisors d of \(R(f_ i,g)\), \(R(\bar f_ i,\overline{h/h_ j})\neq \bar d\) then F is irreducible in K[X]. From this result the criteria for irreducibility of Schönemann, Eisenstein and Akira are derived.
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13A18
zbMATH DE Number: 3995842
zbMATH Keywords: polynomials over valuation rings
zbMATH Keywords: irreducibility
MaRDI profile type: Publication
cites work: Q3966240
cites work: Q4092065
cites work: Q3677831
cites work: Q5666659
description: scientific article; zbMATH DE number 3995842