Factorization of polynomials over valued fields based on graded polynomials

2020 ◽  
Vol 70 (4) ◽  
pp. 807-814
Author(s):  
Lhoussain El Fadil
Keyword(s):  
Prime Ideal ◽  
Newton Polygon ◽  
Monic Polynomial ◽  
New Method ◽  
Valued Fields ◽  
Valued Field ◽  
Hensel’S Lemma ◽  
Factorization Of Polynomials

AbstractIn this paper, we develop a new method based on Newton polygon and graded polynomials, similar to the known one based on Newton polygon and residual polynomials. This new method allows us the factorization of any monic polynomial in any henselian valued field. As applications, we give a new proof of Hensel’s lemma and a theorem on prime ideal factorization.

On Newton polygons techniques and factorization of polynomials over Henselian fields

2019 ◽  
Vol 19 (10) ◽  
pp. 2050188
Author(s):  
Lhoussain El Fadil
Keyword(s):  
Newton Polygon ◽  
Monic Polynomial ◽  
Local Fields ◽  
Algebraic Number Theory ◽  
Newton Polygons ◽  
Valued Field ◽  
Rank One ◽  
Difference Polynomials ◽  
Polynomial Formula ◽  
Factorization Of Polynomials

Let [Formula: see text] be a valued field, where [Formula: see text] is a rank-one discrete valuation, with valuation ring [Formula: see text]. The goal of this paper is to investigate some basic concepts of Newton polygon techniques of a monic polynomial [Formula: see text]; namely, theorem of the product, of the polygon, and of the residual polynomial, in such way that improves that given in [D. Cohen, A. Movahhedi and A. Salinier, Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika 47 (2000) 173–196] and generalizes that given in [J. Guardia, J. Montes and E. Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc. 364(1) (2012) 361–416] to any rank-one valued field.

scholarly journals ON IRREDUCIBLE FACTORS OF POLYNOMIALS OVER COMPLETE FIELDS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250125 ◽  
Author(s):  
SUDESH K. KHANDUJA ◽  
SANJEEV KUMAR
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Manuscripta Math ◽  
Algebraic Number Fields ◽  
Valued Fields ◽  
Valued Field ◽  
Transcendental Extension ◽  
Simple Transcendental Extension ◽  
Irreducibility Criterion ◽  
Hensel’S Lemma

Let (K, v) be a complete rank-1 valued field. In this paper, we extend classical Hensel's Lemma to residually transcendental prolongations of v to a simple transcendental extension K(x) and apply it to prove a generalization of Dedekind's theorem regarding splitting of primes in algebraic number fields. We also deduce an irreducibility criterion for polynomials over rank-1 valued fields which extends already known generalizations of Schönemann Irreducibility Criterion for such fields. A refinement of Generalized Akira criterion proved in Khanduja and Khassa [Manuscripta Math.134(1–2) (2010) 215–224] is also obtained as a corollary of the main result.

Preliminaries

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Galois Coverings ◽  
Definable Type ◽  
Definable Sets ◽  
Background Material

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

Ramification Groups of Abelian Local Field Extensions

1971 ◽  
Vol 23 (2) ◽  
pp. 271-281 ◽  
Author(s):  
Murray A. Marshall
Keyword(s):  
Prime Ideal ◽  
Local Field ◽  
Galois Extension ◽  
Residue Class ◽  
Abelian Extensions ◽  
Ring Of Integers ◽  
Field Extensions ◽  
Valued Field ◽  
Residue Class Field ◽  
Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

GANZSTELLENSÄTZE IN THEORIES OF VALUED FIELDS

2008 ◽  
Vol 08 (01) ◽  
pp. 1-22 ◽  
Author(s):  
DEIRDRE HASKELL ◽  
YOAV YAFFE
Keyword(s):  
Valuation Ring ◽  
Uniform Method ◽  
Algebraic Characterization ◽  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Complete Theories ◽  
The Given ◽  
Relevant Class

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields.

On factorization of polynomials in henselian valued fields

2018 ◽  
Vol 46 (7) ◽  
pp. 3205-3221 ◽  
Author(s):  
Anuj Jakhar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Valued Fields ◽  
Henselian Valued Fields ◽  
Factorization Of Polynomials

scholarly journals NOTES ON EXTREMAL AND TAME VALUED FIELDS

2016 ◽  
Vol 81 (2) ◽  
pp. 400-416
Author(s):  
SYLVY ANSCOMBE ◽  
FRANZ-VIKTOR KUHLMANN
Keyword(s):  
Approximation Property ◽  
Elementary Theory ◽  
Residue Field ◽  
Axiom System ◽  
Complete Characterization ◽  
Valued Fields ◽  
Mixed Characteristic ◽  
Valued Field ◽  
Residue Fields

AbstractWe extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

INVARIANTS OF DEFECTLESS IRREDUCIBLE POLYNOMIALS

2010 ◽  
Vol 09 (04) ◽  
pp. 603-631 ◽  
Author(s):  
RON BROWN ◽  
JONATHAN L. MERZEL
Keyword(s):  
Irreducible Polynomial ◽  
Newton Polygon ◽  
Extension Field ◽  
Irreducible Polynomials ◽  
Technical Result ◽  
Valued Field ◽  
Minimal Pairs ◽  
Simple Characterization ◽  
Polynomial Extensions

Defectless irreducible polynomials over a Henselian valued field (F, v) have been studied by means of strict systems of polynomial extensions and complete (also called "saturated") distinguished chains. Strong connections are developed here between these two approaches and applications made to both. In the tame case in which a root α of an irreducible polynomial f generates a tamely ramified extension of (F, v), simple formulas are given for the Krasner constant, the Brink separant and the diameter of f. In this case a (best possible) result is given showing that a sufficiently good approximation in an extension field K of F to a root of a defectless polynomial f over F guarantees the existence of an exact root of f in K. Also in the tame case a (best possible) result is given describing when a polynomial is sufficiently close to a defectless polynomial so as to guarantee that the roots of the two polynomials generate the same extension fields. Another application in the tame case gives a simple characterization of the minimal pairs (in the sense of N. Popescu et al.). A key technical result is a computation in the tame case of the Newton polygon of f(x+α). Invariants of defectless polynomials are discussed and the existence of defectless polynomials with given invariants is proven. Khanduja's characterization of the tame polynomials whose Krasner constants equal their diameters is generalized to arbitrary defectless polynomials. Much of the work described here will be seen not to require the hypothesis that (F, v) is Henselian.

Un principe d'ax-kochen-ershov pour des structures intermédiates entre groupes et corps valués

1999 ◽  
Vol 64 (3) ◽  
pp. 991-1027 ◽  
Author(s):  
Françoise Delon ◽  
Patrick Simonetta
Keyword(s):  
Abelian Group ◽  
Cross Section ◽  
Additive Group ◽  
Ordered Group ◽  
Unary Predicate ◽  
Valued Fields ◽  
Valued Field ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Image Position

AbstractAn Ax-Kochen-Ershov principle for intermediate structures between valued groups and valued fields.We will consider structures that we call valued B-groups and which are of the form 〈G, B, *, υ〉 where– G is an abelian group,– B is an ordered group,– υ is a valuation denned on G taking its values in B,– * is an action of B on G satisfying: ∀x ϵ G ∀ b ∈ B υ(x * b) = ν(x) · b.The analysis of Kaplanski for valued fields can be adapted to our context and allows us to formulate an Ax-Kochen-Ershov principle for valued B-groups: we axiomatise those which are in some sense existentially closed and also obtain many of their model-theoretical properties. Let us mention some applications:1. Assume that υ(x) = υ(nx) for every integer n ≠ 0 and x ϵ G, B is solvable and acts on G in such a way that, for the induced action, Z[B] ∖ {0} embeds in the automorphism group of G. Then 〈G, B, *, υ〉 is decidable if and only if B is decidable as an ordered group.2. Given a field k and an ordered group B, we consider the generalised power series field k((B)) endowed with its canonical valuation. We consider also the following structure:where k((B))+ is the additive group of k((B)), S is a unary predicate interpreting {Tb ∣ b ϵB}, and ×↾k((B))×S is the multiplication restricted to k((B)) × S, structure which is a reduct of the valued field k((B)) with its canonical cross section. Then our result implies that if B is solvable and decidable as an ordered group, then M is decidable.3. A valued B–group has a residual group and our Ax-Kochen-Ershov principle remains valid in the context of expansions of residual group and value group. In particular, by adding a residual order we obtain new examples of solvable ordered groups having a decidable theory.

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