Simultaneous approximation to algebraic numbers by elements of a number field

1975 ◽  
Vol 79 (1) ◽  
pp. 55-66 ◽  
Author(s):  
Wolfgang M. Schmidt
Keyword(s):  
Number Field ◽  
Simultaneous Approximation ◽  
Algebraic Numbers

scholarly journals Kummer theory for number fields and the reductions of algebraic numbers

2019 ◽  
Vol 15 (08) ◽  
pp. 1617-1633 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba
Keyword(s):  
Number Field ◽  
Arithmetic Progression ◽  
Direct Proof ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Kummer Theory ◽  
Multiplicative Order ◽  
Higher Rank ◽  
Multiplicative Subgroup ◽  
Torsion Free

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if [Formula: see text] is a finitely generated and torsion-free multiplicative subgroup of a number field [Formula: see text] having rank [Formula: see text], then the ratio between [Formula: see text] and the Kummer degree [Formula: see text] is bounded independently of [Formula: see text]. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

scholarly journals Simultaneous approximation of et and ℘(t)

1982 ◽  
Vol 33 (2) ◽  
pp. 171-178
Author(s):  
K. Saradha
Keyword(s):  
Complex Number ◽  
Elliptic Function ◽  
Simultaneous Approximation ◽  
Algebraic Numbers ◽  
Weierstrass Elliptic Function ◽  
Algebraic Invariants ◽  
Image Position

AbstractLet t be any complex number different from the poles of a Weierstrass elliptic function ℘(z), having algebraic invariants. Then we estimate from below the sum where α and β are algebraic numbers. The estimate is given in terms of the heights of α and β and the degree of the field Q(α, β), where Q is the field of rationals.

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.

Simultaneous rational approximations to certain algebraic numbers

1967 ◽  
Vol 63 (3) ◽  
pp. 693-702 ◽  
Author(s):  
A. Baker
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Infinitely Many Solutions ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Famous Theorem ◽  
Image Position

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

scholarly journals IRRATIONALITY AND NONQUADRATICITY MEASURES FOR LOGARITHMS OF ALGEBRAIC NUMBERS

2012 ◽  
Vol 92 (2) ◽  
pp. 237-267 ◽  
Author(s):  
RAFFAELE MARCOVECCHIO ◽  
CARLO VIOLA
Keyword(s):  
Saddle Point ◽  
Number Field ◽  
Point Method ◽  
Saddle Point Method ◽  
Algebraic Numbers ◽  
Double Complex

AbstractLet 𝕂⊂ℂ be a number field. We show how to compute 𝕂-irrationality measures of a number ξ∉𝕂, and 𝕂-nonquadraticity measures of ξ if [𝕂(ξ):𝕂]>2. By applying the saddle point method to a family of double complex integrals, we prove ℚ(α)-irrationality measures and ℚ(α)-nonquadraticity measures of log α for several algebraic numbers α∈ℂ, improving earlier results due to Amoroso and the second-named author.

scholarly journals ON NUMBER FIELDS WITH NONTRIVIAL SUBFIELDS

2011 ◽  
Vol 07 (03) ◽  
pp. 695-720 ◽  
Author(s):  
MARTIN WIDMER
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Interpretation Of Probability

What is the probability for a number field of composite degree d to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if d > 6. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this paper is an estimate for the number of algebraic numbers of degree d = en and bounded height which generate a field that contains an unspecified subfield of degree e. If n > max {e2 + e, 10}, we get the correct asymptotics as the height tends to infinity.

scholarly journals Groups of Matrices With Integer Eigenvalues

1971 ◽  
Vol 12 (3) ◽  
pp. 351-357 ◽  
Author(s):  
M. R. Freislich
Keyword(s):  
Number Field ◽  
Characteristic Polynomial ◽  
Linear Group ◽  
Algebraic Number ◽  
General Linear Group ◽  
Algebraic Number Field ◽  
General Linear ◽  
Algebraic Numbers ◽  
Algebraic Integers

Let F be an algebraic number field, and S a subgroup of the general linear group GL(n, F). We shall call S a U-group if S satisfies the condition (U): Every x ∈ S is a matrix all of whose eigenvalues are algebraic integers. (This is equivalent to either of the following conditions: a) the eigenvalues of each matrix (x are all units as algebraic numbers; b) the characteristic polynomial for x has all its coefficients integers in F. In particular, then, every group of matrices with entries in the integers of F is a U-group.

Simultaneous approximation of numbers connected with the exponential function

1978 ◽  
Vol 25 (4) ◽  
pp. 466-478 ◽  
Author(s):  
Michel Waldschmidt
Keyword(s):  
Lower Bounds ◽  
Exponential Function ◽  
Complex Number ◽  
Simultaneous Approximation ◽  
Complex Numbers ◽  
Linear Forms In Logarithms ◽  
Algebraic Numbers ◽  
Linear Forms

AbstractWe give several results concerning the simultaneous approximation of certain complex numbers. For instance, we give lower bounds for |a–ξo |+ | ea – ξ1 |, where a is any non-zero complex number, and ξ are two algebraic numbers. We also improve the estimate of the so-called Franklin Schneider theorem concerning | b – ξ | + | a – ξ | + | ab – ξ. We deduce these results from an estimate for linear forms in logarithms.

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