# 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

$P(x)=\mathrm{\hspace{0.33em}0}$ | (1) |

where

$$P(x):={a}_{0}{x}^{n}+{a}_{1}{x}^{n-1}+\mathrm{\dots}+{a}_{n-1}x+{a}_{n}$$ |

is an irreducible (http://planetmath.org/IrreduciblePolynomial2) and primitive polynomial with integer coefficients ${a}_{j}$ and ${a}_{0}>0$. Each algebraic number exactly one such equation (see the minimal polynomial). For every integer $N=2,\mathrm{\hspace{0.17em}3},\mathrm{\hspace{0.17em}4},\mathrm{\dots}$ there exists a finite number of equations (1) such that

$$n+{a}_{0}+|{a}_{1}|+\mathrm{\dots}+|{a}_{n}|=N$$ |

(e.g. if $N=3$, then one has the equations $x-1=0$ and
$x+1=0$) 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) ${S}_{N}$ using a system, for example by the magnitude of the real part^{} and the imaginary part. When one forms the concatenated sequence

$${S}_{2},{S}_{3},{S}_{4},\mathrm{\dots}$$ |

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

## 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 |