# algebraic number

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Related to algebraic number: Algebraic number field, Algebraic number theory

## algebraic number:

see numbernumber,
entity describing the magnitude or position of a mathematical object or extensions of these concepts. The Natural Numbers

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of objects if their
.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Algebraic Number

any number ɑ satisfying the algebraic equation a1 an + . . . + aka a + an+1 = 0, where n ≥ 1 and a1, . . . , an and an+1 are whole (rational) numbers. The number ɑ is called an algebraic integer if a1 = 1. If the polynomial f(x) = a1xn+ . . . + anx + an+1 is not the product of two other polynomials of positive degree with rational coefficients, then the number n is called the degree of the algebraic number a. The simplest algebraic numbers are the roots of the binomial equation xn = a, where α is a rational number. For example, algebraic numbers are the rational numbers and the numbers integral algebraic numbers are the integers and the numbers . Two important trends in number theory are closely associated with the concept of the algebraic number. The first is the arithmetic of algebraic numbers (algebraic theory of numbers), originated by E. Kummer in the middle of the 19th century, which studies properties of algebraic numbers. Algebraic integers possess a number of properties analogous to those of integral rational numbers; however, the unique factorization theorem has no place in the theory of algebraic integers. To preserve unique factorization, Kummer introduced his so-called ideal numbers. The second trend is the theory of approximation of algebraic numbers, which studies the degree of approximation of these numbers by rational numbers or by algebraic numbers themselves. The first result in this direction was the theorem of J. Liouville, which showed that algebraic numbers are “badly” approximated by rational numbers. More precisely, if α is an algebraic number of degree n, then, for any integral rational values of ρ and q, the inequality ǀa − p/q ǀ > C/qn, holds, where C = C(a) > 0 is a constant independent of ρ and q, from which it follows that it is not difficult to construct an arbitrary number of nonalgebraic numbers—the transcendental numbers.

### REFERENCES

Gekke, E. Lektsii po teorii algebraicheskikh chisel. Moscow-Leningrad, 1940. (Translated from German.)
Gel’fond, A. O. Transtsendentnye i algebraicheskie chista. Moscow, 1952.
Borevich, Z. I., and I. R. Shafarevich. Teoriia chisel. Moscow, 1964.

A. A. KARATSUBA

## algebraic number

[¦al·jə¦brā·ik ′nəm·bər]
(mathematics)
Any root of a polynomial with rational coefficients.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
See, e.g., [2] for the description of all algebraic numbers expressible as differences or as quotients of two conjugate algebraic numbers and [3], [7] for some further work on this problem.
Although our model is very general and allows to compute a large set of numbers, some algebraic numbers as "simple" (on a computational point of view) as 1/5 are not computable.
Let a, b, c > 0 and u be a real algebraic number. For x [member of] R, n [greater than or equal to] 0 and For any nonnegative integer l and real numbers [alpha] and [beta] we have
(i) The usual definition is more general, the coefficients of S can be taken in an arbitrary algebraic number field.
In this paper, we study the remaining parts, namely, we give such bounds for modular forms with Fourier coefficients in an arbitrary algebraic number field K and for any prime ideal p in K.
Other topics are rational points on elliptic curves, conics and the p-adic numbers, the zeta function, and algebraic number theory.
After a fine introduction to basic notions, he covers unique factorization, the Gaussian integers, and Pell's equation, and moves on to algebraic number theory.
354] proved that the compactness of [J.sup.1.sub.F] is equivalent to the two basic finiteness results of algebraic number theory: (i) Pic([O.sub.F]) is finite; (ii) [O.sup.x.sub.F] is finitely generated.
The 1882 Theorie der algebraischen Functionen einer Veranderlichen by Dedekind (1831-1916) and Weber (1842-1913) changed the direction of algebraic geometry and established its foundations by introducing methods from algebraic number theory.
Mathematics Institute devoted to the connection between algebraic number theory and low-dimensional topology, this volume contains 15 papers discussing analogies between primes and knots, number rings, and 3- manifolds; avant-garde Galois theory (Galois representations and Grothendieck-Teichmuller groups); classical and quantum topology of knots; and arithmetic non-commutative geometry and related mathematical physics.
Prerequisites are algebraic geometry, algebraic number theory, and some group cohomology.
Then using Lemma 2.2 we see that the algebraic number g([k.sub.1]) - g([k.sub.2]) is expressed as

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