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Algebraic Numbers [PDF]

Formalized Mathematics, 2016
Summary This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”.
Yasushige Watase
semanticscholar   +4 more sources

Frieze patterns over algebraic numbers [PDF]

Bulletin of the London Mathematical Society, 2023
Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jørgensen ...
M. Cuntz, T. Holm, Carlo Pagano
semanticscholar   +1 more source

Algebraic values of sines and cosines and their arguments

Lietuvos Matematikos Rinkinys, 2021
The article introduces the reader to some amazing properties of trigonometric functions. It turns out that if the values of the arguments of the functions sin x, cos x, tg x and ctg x, expressed in radians, are algebraic numbers, then the values of these
Edmundas Mazėtis, Grigorijus Melničenko   +1 more
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Fractional parts of powers of real algebraic numbers

Comptes Rendus. Mathématique, 2022
Let $\alpha $ be a real algebraic number greater than $1$. We establish an effective lower bound for the distance between an integral power of $\alpha $ and its nearest integer.
Bugeaud, Yann
doaj   +1 more source

A finite analogue of the ring of algebraic numbers [PDF]

Journal of Number Theory, 2018
We construct an analogue of the ring of algebraic numbers, living in a quotient of the product of all finite fields of prime order. We use this ring to deduce some results about linear recurrent sequences.
Julian Rosen
semanticscholar   +1 more source

An Effective Framework for Constructing Exponent Lattice Basis of Nonzero Algebraic Numbers [PDF]

International Symposium on Symbolic and Algebraic Computation, 2018
Computing a basis for the exponent lattice of algebraic numbers is a basic problem in the field of computational number theory with applications to many other areas.
Taotao Zheng, Bican Xia
semanticscholar   +1 more source

Algebraic Numbers of the form α^T with α Algebraic and T Transcendental

Mathematics, 2020
Let α≠1 be a positive real number and let P(x) be a non-constant rational function with algebraic coefficients. In this paper, in particular, we prove that the set of algebraic numbers of the form αP(T), with T transcendental, is dense in some open ...
Štěpán Hubálovský, Eva Trojovská
doaj   +1 more source

Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination [PDF]

Logical Methods in Computer Science, 2012
This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties.
Assia Mahboubi, Cyril Cohen
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On the Degree of Product of Two Algebraic Numbers

Mathematics, 2023
A triplet (a,b,c) of positive integers is said to be product-feasible if there exist algebraic numbers α, β and γ of degrees (over Q) a, b and c, respectively, such that αβγ=1.
Lukas Maciulevičius
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Denumerability of the Algebraic Numbers

Davao Research Journal, 2000
An algebraic number is a real number that is a root of a polynomial equation anXn + an-1Xn-1…+a0 where ai are integers. In this paper, using the fact that a polynomial equation of degree n has at most n roots, together with some results, the ...
Marleonie Bauyot
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