scholarly journals On the Irreducible Factors of a Polynomial and Applications to Extensions of Absolute Values

2021 ◽  
Author(s):  
Lhoussain El Fadil ◽  
Mohamed Faris
Keyword(s):  
Algebraic Number Theory ◽  
Sufficient Condition ◽  
Polynomial Factorization ◽  
General Version ◽  
Field Extensions ◽  
Valued Field ◽  
Rank One ◽  
Key Polynomials ◽  
Irreducibility Criterion

Polynomial factorization over a field is very useful in algebraic number theory, in extensions of valuations, etc. For valued field extensions, the determination of irreducible polynomials was the focus of interest of many authors. In 1850, Eisenstein gave one of the most popular criterion to decide on irreducibility of a polynomial over Q. A criterion which was generalized in 1906 by Dumas. In 2008, R. Brown gave what is known to be the most general version of Eisenstein-Schönemann irreducibility criterion. Thanks to MacLane theory, key polynomials play a key role to extend absolute values. In this chapter, we give a sufficient condition on any monic plynomial to be a key polynomial of an absolute value, an irreducibly criterion will be given, and for any simple algebraic extension L=Kα, we give a method to describe all absolute values of L extending ∣∣, where K is a discrete rank one valued field.

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.

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.

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.

scholarly journals Counting the number of distinct distances of elements in valued field extensions

2018 ◽  
Vol 509 ◽  
pp. 192-211 ◽  
Author(s):  
Anna Blaszczok ◽  
Franz-Viktor Kuhlmann
Keyword(s):  
Field Extensions ◽  
Valued Field

scholarly journals An application of bivariate polynomial factorization on decoding of Reed-Solomon based codes

2018 ◽  
Vol 12 (1) ◽  
pp. 166-177
Author(s):  
Ivan Pavkov ◽  
Nebojsa Ralevic ◽  
Ljubo Nedovic
Keyword(s):  
Efficient Algorithm ◽  
Sufficient Condition ◽  
Polynomial Factorization ◽  
Necessary And Sufficient Condition ◽  
Bivariate Polynomials ◽  
Bivariate Polynomial ◽  
Integer Coefficients ◽  
Necessary And Sufficient ◽  
The Given

A necessary and sufficient condition for the existence of a non-trivial factorization of an arbitrary bivariate polynomial with integer coefficients was presented in [2]. In this paper we develop an efficient algorithm for factoring bivariate polynomials with integer coefficients. Also, we shall give a proof of the optimality of the algorithm. For a given codeword, formed by mixing up two codewords, the algorithm recovers those codewords directly by factoring corresponding bivariate polynomial. Our algorithm determines uniquely the given polynomials which are used in forming the mixture of two codewords.

5.—The Uniform Asymptotic Stability of Certain Linear Differential-difference Equations

1976 ◽  
Vol 74 ◽  
pp. 71-79
Author(s):  
R. Datko
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.

Determination of Grassmann Manifolds Which are Boundaries

1991 ◽  
Vol 34 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Parameswaran Sankaran
Keyword(s):  
Grassmann Manifold ◽  
Vector Subspace ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Grassmann Manifolds

AbstractLetFGn,kdenote the Grassmann manifold of allk-dimensional (left)F-vector subspace ofFnfor F = ℝ, the reals,C, the complex numbers, orHthe quaternions. The problem of determining which of the Grassmannians bound was addressed by the author in [4]. Partial results were obtained in [4] for the caseF= ℝ, including a sufficient condition, due to A. Dold, on n and k for ℝGn,kto bound. Here, we show that Dold's condition is also necessary, and obtain a new proof of sufficiency using the methods of this paper, which cover the complex and quaternionic cases as well.

On the Modulii of Analytic Functions

1970 ◽  
Vol 13 (3) ◽  
pp. 325-327 ◽  
Author(s):  
Malcolm J. Sherman
Keyword(s):  
Analytic Functions ◽  
Point Of View ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Necessary And Sufficient Condition ◽  
Concrete Form ◽  
Norm Equality ◽  
Image Position ◽  
Necessary And Sufficient

The problem to be considered in this note, in its most concrete form, is the determination of all quartets f1, f2, g1, g2 of functions analytic on some domain and satisfying*where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.

scholarly journals The computation of stationary distributions of Markov chains through perturbations

1991 ◽  
Vol 4 (1) ◽  
pp. 29-46 ◽  
Author(s):  
Jeffery J. Hunter
Keyword(s):  
Stochastic Matrix ◽  
Linear Equations ◽  
Probability Vector ◽  
Doubly Stochastic ◽  
Rank One ◽  
Systems Of Linear Equations ◽  
Stationary Probability Vector ◽  
Irreducible Markov Chain ◽  
Algorithmic Procedure

An algorithmic procedure for the determination of the stationary distribution of a finite, m-state, irreducible Markov chain, that does not require the use of methods for solving systems of linear equations, is presented. The technique is based upon a succession of m, rank one, perturbations of the trivial doubly stochastic matrix whose known steady state vector is updated at each stage to yield the required stationary probability vector.

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