algebraic numbers are countable


The set of (a) all algebraic numbersMathworldPlanetmath, (b) the real algebraic numbers is countableMathworldPlanetmath.

Proof.  Let’s consider the algebraic equations

P(x)= 0 (1)



is an irreducible ( and primitive polynomial with integer coefficients aj and  a0>0.  Each algebraic number 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


(e.g. if  N=3,  then one has the equations  x-1=0  and  x+1=0) and thus only a finite setMathworldPlanetmath of algebraic numbers as the of these equations.  These algebraic numbers may be ordered to a finite sequencePlanetmathPlanetmath ( SN using a system, for example by the magnitude of the real partMathworldPlanetmath and the imaginary part.  When one forms the concatenated sequence


it comprises all algebraic numbers in a countable setting, which defines a bijection from the set onto +.


  • 1 E. Kamke: Mengenlehre.  Sammlung Göschen: Band 999/999a.  – Walter de Gruyter & Co., Berlin (1962).
Title algebraic numbers are countable
Canonical name AlgebraicNumbersAreCountable
Date of creation 2013-03-22 15:13:47
Last modified on 2013-03-22 15:13:47
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Theorem
Classification msc 11R04
Classification msc 03E10
Synonym algebraic numbers may be set in a sequence
Related topic HeightOfAnAlgebraicNumber2
Related topic ProofOfTheExistenceOfTranscendentalNumbers
Related topic A_nAreCountableSoIsA_1XXA_nIfA_1
Related topic ExamplesOfCountableSets
Related topic FieldOfAlgebraicNumbers