scholarly journals ON SUBFIELDS OF A FIELD GENERATED BY TWO CONJUGATE ALGEBRAIC NUMBERS

2004 ◽  
Vol 47 (1) ◽  
pp. 119-123 ◽  
Author(s):  
Paulius Drungilas ◽  
Artūras Dubickas
Keyword(s):  
Algebraic Number ◽  
Monic Polynomial ◽  
Subject Classification ◽  
Real Field ◽  
Algebraic Numbers ◽  
Root Of Unity ◽  
Conjugate Algebraic Numbers

AbstractLet $k$ be a field, and let $\alpha$ and $\alpha'$ be two algebraic numbers conjugate over $k$. We prove a result which implies that if $L\subset k(\alpha,\alpha')$ is an abelian or Hamiltonian extension of $k$, then $[L:k]\leq[k(\alpha):k]$. This is related to a certain question concerning the degree of an algebraic number and the degree of a quotient of its two conjugates provided that the quotient is a root of unity, which was raised (and answered) earlier by Cantor. Moreover, we introduce a new notion of the non-torsion power of an algebraic number and prove that a monic polynomial in $X$—irreducible over a real field and having $m$ roots of equal modulus, at least one of which is real—is a polynomial in $X^m$.AMS 2000 Mathematics subject classification: Primary 11R04; 11R20; 11R32; 12F10

scholarly journals SIMULTANEOUS APPROXIMATION BY CONJUGATE ALGEBRAIC NUMBERS IN FIELDS OF TRANSCENDENCE DEGREE ONE

2005 ◽  
Vol 01 (03) ◽  
pp. 357-382 ◽  
Author(s):  
DAMIEN ROY
Keyword(s):  
Simultaneous Approximation ◽  
Transcendence Degree ◽  
Algebraic Numbers ◽  
Bounded Degree ◽  
Root Of Unity ◽  
Transcendental Numbers ◽  
Algebraic Difference ◽  
Conjugate Algebraic Numbers ◽  
The Given ◽  
Real Complex

We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers ξ1, …, ξt by conjugate algebraic numbers of bounded degree over ℚ, provided that the given transcendental numbers ξ1, …, ξt generate over ℚ a field of transcendence degree one. We provide sharper estimates for example when ξ1, …, ξt form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence criterion expressed in terms of polynomials of bounded degree taking small values at ξ1, …, ξt.

scholarly journals Remarks on the approximation to an algebraic number by algebraic numbers.

1986 ◽  
Vol 33 (1) ◽  
pp. 83-93 ◽  
Author(s):  
E. Bombieri ◽  
J. Mueller
Keyword(s):  
Algebraic Number ◽  
Algebraic Numbers

scholarly journals Joint distribution of conjugate algebraic numbers: A random polynomial approach

2020 ◽  
Vol 359 ◽  
pp. 106849 ◽  
Author(s):  
Friedrich Götze ◽  
Denis Koleda ◽  
Dmitry Zaporozhets
Keyword(s):  
Joint Distribution ◽  
Random Polynomial ◽  
Algebraic Numbers ◽  
Conjugate Algebraic Numbers

Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point

1997 ◽  
Vol 49 (5) ◽  
pp. 887-915 ◽  
Author(s):  
Peter Borwein ◽  
Christopher Pinner
Keyword(s):  
Algebraic Number ◽  
Real Root ◽  
Small Degree ◽  
Pisot Number ◽  
Roots Of Unity ◽  
Best Approximations ◽  
Rate Of Approximation ◽  
Root Of Unity ◽  
Extremal Polynomials ◽  
Image Position

AbstractFor a fixed algebraic number α we discuss how closely α can be approximated by a root of a {0, +1, -1} polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, k, of the polynomial at α.In particular we obtain the following. Let BN denote the set of roots of all {0, +1, -1} polynomials of degree at most N and BN(α k) the roots of those polynomials that have a root of order at most k at α. For a Pisot number α in (1, 2] we show thatand for a root of unity α thatWe study in detail the case of α = 1, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When k = 0 or 1 we can describe the extremal polynomials explicitly.

scholarly journals Algebraic Numbers Satisfying Polynomials with Positive Rational Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Vichian Laohakosol ◽  
Suton Tadee
Keyword(s):  
Growth Factor ◽  
Real Number ◽  
Algebraic Number ◽  
Positive Real Number ◽  
Algebraic Numbers ◽  
Positive Real ◽  
Quantitative Version

A theorem of Dubickas, affirming a conjecture of Kuba, states that a nonzero algebraic number is a root of a polynomial f with positive rational coefficients if and only if none of its conjugates is a positive real number. A certain quantitative version of this result, yielding a growth factor for the coefficients of f similar to the condition of the classical Eneström-Kakeya theorem of such polynomial, is derived. The bound for the growth factor so obtained is shown to be sharp for some particular classes of algebraic numbers.

scholarly journals On real algebraic numbers in which the derivative of their minimal polynomial is small

2021 ◽  
Vol 57 (2) ◽  
pp. 135-147
Author(s):  
D. V. Koleda
Keyword(s):  
Algebraic Number ◽  
Minimal Polynomial ◽  
Algebraic Numbers ◽  
Absolute Value ◽  
Positive Degree ◽  
Integer Polynomials ◽  
The Absolute

Algebraic numbers are the roots of integer polynomials. Each algebraic number α is characterized by its minimal polynomial Pα that is a polynomial of minimal positive degree with integer coprime coefficients, α being its root. The degree of α is the degree of this polynomial, and the height of α is the maximum of the absolute values of the coefficients of this polynomial. In this paper we consider the distribution of algebraic numbers α whose degree is fixed and height bounded by a growing parameter Q, and the minimal polynomial Pα is such that the absolute value of its derivative P'α (α) is bounded by a given parameter X. We show that if this bounding parameter X is from a certain range, then as Q → +∞ these algebraic numbers are distributed uniformly in the segment [-1+√2/3.1-√2/3]

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