scholarly journals Roots of unity as quotients of two conjugate algebraic numbers

2017 ◽  
Vol 52 (2) ◽  
pp. 235-240
Author(s):  
Artūras Dubickas ◽  
Keyword(s):  
Roots Of Unity ◽  
Algebraic Numbers ◽  
Conjugate 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

ROOTS OF POLYNOMIALS WITH DOMINANT TERM

2011 ◽  
Vol 07 (05) ◽  
pp. 1217-1228 ◽  
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Constant Coefficient ◽  
Roots Of Unity ◽  
Algebraic Numbers ◽  
Algebraic Integers ◽  
Dominant Term ◽  
Minimal Weight ◽  
Number Systems ◽  
Integer Polynomials ◽  
Roots Of Polynomials ◽  
Integer Polynomial

We characterize all algebraic numbers which are roots of integer polynomials with a coefficient whose modulus is greater than or equal to the sum of moduli of all the remaining coefficients. It turns out that these numbers are zero, roots of unity and those algebraic numbers β whose conjugates over ℚ (including β itself) do not lie on the circle |z| = 1. We also describe all algebraic integers with norm B which are roots of an integer polynomial with constant coefficient B and the sum of moduli of all other coefficients at most |B|. In contrast to the above, the set of such algebraic integers is "quite small". These results are motivated by a recent paper of Frougny and Steiner on the so-called minimal weight β-expansions and are also related to some work on canonical number systems and tilings.

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 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

Distance between conjugate algebraic numbers in clusters

2013 ◽  
Vol 94 (5-6) ◽  
pp. 816-819 ◽  
Author(s):  
N. V. Budarina ◽  
F. Goetze
Keyword(s):  
Algebraic Numbers ◽  
Conjugate Algebraic Numbers

scholarly journals Conjugate algebraic numbers on circles

1976 ◽  
Vol 29 (2) ◽  
pp. 147-157
Author(s):  
Veikko Ennola ◽  
C. Smyth
Keyword(s):  
Algebraic Numbers ◽  
Conjugate Algebraic Numbers
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