algebraic numbers are countable
Theorem.
The set of (a) all algebraic numbers, (b) the real algebraic numbers is countable.
Proof. Let’s consider the algebraic equations
(1) |
where
is an irreducible (http://planetmath.org/IrreduciblePolynomial2) and primitive polynomial with integer coefficients and . Each algebraic number exactly one such equation (see the minimal polynomial). For every integer there exists a finite number of equations (1) such that
(e.g. if , then one has the equations and ) and thus only a finite set of algebraic numbers as the of these equations. These algebraic numbers may be ordered to a finite sequence (http://planetmath.org/OrderedTuplet) using a system, for example by the magnitude of the real part 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 .
References
- 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 |