Algebraic Numbers and Number Fields

1998 ◽  
pp. 3-7
Author(s):  
David Hilbert
Keyword(s):  
Number Fields ◽  
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).

Espaces adéliques quadratiques

2016 ◽  
Vol 162 (2) ◽  
pp. 211-247 ◽  
Author(s):  
ÉRIC GAUDRON ◽  
GAËL RÉMOND
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Quadratic Forms ◽  
Number Fields ◽  
Orthogonal Basis ◽  
Ambient Space ◽  
Algebraic Extension ◽  
Isotropic Subspace ◽  
Algebraic Numbers ◽  
Isotropic Vectors

AbstractWe study quadratic forms defined on an adelic vector space over an algebraic extension K of the rationals. Under the sole condition that a Siegel lemma holds over K, we provide height bounds for several objects naturally associated to the quadratic form, such as an isotropic subspace, a basis of isotropic vectors (when it exists) or an orthogonal basis. Our bounds involve the heights of the form and of the ambient space. In several cases, we show that the exponents of these heights are best possible. The results improve and extend previously known statements for number fields and the field of algebraic numbers.

scholarly journals Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

2020 ◽  
Vol 15 (1) ◽  
pp. 75-92 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba
Keyword(s):  
Rational Number ◽  
Prime Power ◽  
Number Fields ◽  
Free Subgroup ◽  
Algebraic Numbers ◽  
Kummer Theory ◽  
Adic Valuation ◽  
Torsion Free Subgroup ◽  
Almost All ◽  
Torsion Free

AbstractLet K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

scholarly journals Singular values of multiple eta-quotients for ramified primes

2013 ◽  
Vol 16 ◽  
pp. 407-418 ◽  
Author(s):  
Andreas Enge ◽  
Reinhard Schertz
Keyword(s):  
Complex Multiplication ◽  
Singular Values ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Modular Functions ◽  
Class Invariants ◽  
Free Level ◽  
Yield Class ◽  
Imaginary Quadratic Number ◽  
Class Fields

AbstractWe determine the conditions under which singular values of multiple $\eta $-quotients of square-free level, not necessarily prime to six, yield class invariants; that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index ${2}^{{k}^{\prime } - 1} $ when ${k}^{\prime } \geq 2$ primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on ${ X}_{0}^{+ } (p)$ for $p$ prime and ramified.

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 Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations

2022 ◽  
Vol 6 (1) ◽  
pp. 39
Author(s):  
Christoph Bandt ◽  
Dmitry Mekhontsev
Keyword(s):  
Fractal Geometry ◽  
Number Fields ◽  
Computer Assisted ◽  
Algebraic Numbers ◽  
Open Set Condition ◽  
Symmetry Classes ◽  
Open Set ◽  
Strong Connectedness ◽  
Self Similar

Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields.

Multiplicative relations in number fields

1977 ◽  
Vol 16 (1) ◽  
pp. 83-98 ◽  
Author(s):  
A.J. van der Poorten ◽  
J.H. Loxton
Keyword(s):  
Explicit Form ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Linear Forms ◽  
Independence Condition

In this paper, we obtain an explicit form of the currently best known inequality for linear forms in the logarithms of algebraic numbers. The results complete our previous investigations (Bull. Austral. Math. Soc. 15 (1976), 33–57) which were conditional on a certain independence condition on the algebraic numbers. The extra work needed to obtain unconditional results centres on the properties of multiplicative relations in number fields. In particular, we show that a set of multiplicatively dependent algebraic numbers always satisfies a relation with relatively small exponents.

scholarly journals Minimal degrees of algebraic numbers with respect to primitive elements

2021 ◽  
pp. 1-16
Author(s):  
Cheol-Min Park ◽  
Sun Woo Park
Keyword(s):  
Algebraic Number ◽  
Primitive Element ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Primitive Elements ◽  
Arithmetic Properties ◽  
Congruent Number ◽  
Number Problem ◽  
Number Formula ◽  
Congruent Number Problem

Given a number field [Formula: see text], we define the degree of an algebraic number [Formula: see text] with respect to a choice of a primitive element of [Formula: see text]. We propose the question of computing the minimal degrees of algebraic numbers in [Formula: see text], and examine these values in degree 4 Galois extensions over [Formula: see text] and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.

scholarly journals Constructing totally p-adic numbers of small height

2021 ◽  
pp. 1-14
Author(s):  
S. Checcoli ◽  
A. Fehm
Keyword(s):  
Upper Bound ◽  
Galois Extension ◽  
Number Fields ◽  
Prime Numbers ◽  
Local Behavior ◽  
Algebraic Numbers ◽  
Galois Extensions ◽  
Small Height ◽  
Finite Set

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behavior at finitely many places.

scholarly journals EQUIDISTRIBUTION OF ALGEBRAIC NUMBERS OF NORM ONE IN QUADRATIC NUMBER FIELDS

2011 ◽  
Vol 07 (07) ◽  
pp. 1841-1861
Author(s):  
KATHLEEN L. PETERSEN ◽  
CHRISTOPHER D. SINCLAIR
Keyword(s):  
Unit Circle ◽  
Quadratic Extension ◽  
Number Fields ◽  
Quadratic Number ◽  
Algebraic Numbers ◽  
Minimal Norm ◽  
Weil Height ◽  
Distribution Of Elements ◽  
Hilbert’S Theorem 90 ◽  
Real Quadratic

Given a fixed quadratic extension K of ℚ, we consider the distribution of elements in K of norm one (denoted [Formula: see text]). When K is an imaginary quadratic extension, [Formula: see text] is naturally embedded in the unit circle in ℂ and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in [Formula: see text] can be written as [Formula: see text] for some [Formula: see text], which yields another ordering of [Formula: see text] given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that [Formula: see text] is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that [Formula: see text] is equidistributed with respect to norm, under the map β ↦ log |β|( mod log |ϵ2|) where ϵ is a fundamental unit of [Formula: see text].

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