Algebraic Number (redirected from Algebraic numbers)

algebraic number: see number.

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algebraic number [¦al·jə¦brā·ik ′nəm·bər]

(mathematics)

Any root of a polynomial with rational coefficients.

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Algebraic Number 

any number ɑ satisfying the algebraic equation a₁ aⁿ + . . . + a_ka a + a_n+1 = 0, where n ≥ 1 and a₁, . . . , a_n and a_n+1 are whole (rational) numbers. The number ɑ is called an algebraic integer if a₁ = 1. If the polynomial f(x) = a₁xⁿ+ . . . + a_nx + a_n+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/q^n, 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