PlanetMath: algebraic numbers are countable

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algebraic numbers are countable

(Theorem)

Theorem 1 The set of (a) all algebraic numbers, (b) the real algebraic numbers is countable.

Proof. Let's consider the algebraic equations

$\displaystyle P(x) = 0$

(1)

where

$\displaystyle P(x) := a_0x^n\!+\!a_1x^{n-1}\!+\!\cdots\!+\!a_{n-1}x\!+\!a_n$

is an irreducible and primitive polynomial with integer coefficients

and

. Each algebraic number satisfies exactly one such equation (see the minimal polynomial). For every integer $N = 2,\,3,\,4,\,...$ there exists a finite number of equations (1) such that

$\displaystyle n\!+\!a_0\!+\!\vert a_1\vert\!+\cdots+\!\vert a_n\vert = N$

(e.g. if

, then one has the equations $x\!-\!1 = 0$ and $x\!+\!1 = 0$ ) and thus only a finite set of algebraic numbers as the roots of these equations. These algebraic numbers may be ordered to a finite sequence

using a fixed ordering system, for example by the magnitude of the real part and the imaginary part. When one forms the concatenated sequence

$\displaystyle S_2,\,S_3,\,S_4,\,...$

it comprises all algebraic numbers in a countable setting, which defines a bijection from the set onto $\mathbb{Z}_+$ .

Bibliography

1: E. KAMKE: Mengenlehre. Sammlung Göschen: Band 999/999a. - Walter de Gruyter & Co., Berlin (1962).

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algebraic numbers may be set in a sequence

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Attachments:: all algebraic numbers in a sequence (Result) by pahio

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Cross-references: onto, bijection, sequence, imaginary part, real part, finite set, number, finite, minimal polynomial, equation, coefficients, integer, primitive polynomial, algebraic equations, countable, real, algebraic numbers
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This is version 10 of algebraic numbers are countable, born on 2005-05-02, modified 2006-09-08.
Object id is 6999, canonical name is AlgebraicNumbersAreCountable.
Accessed 8021 times total.

Classification:

AMS MSC:	03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
	11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)

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