PlanetMath: algebraic numbers are countable

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

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).

"algebraic numbers are countable" is owned by pahio.

(view preamble)

Other names:

algebraic numbers may be set in a sequence

This object's parent.

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

Log in to rate this entry.
(view current ratings)

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
There are 4 references to this entry.

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 8020 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)

Pending Errata and Addenda

None.[ View all 2 ]

Discussion

forum policy

No messages.

Interact