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

 $\displaystyle P(x)\;=\;0$ (1)

where

 $P(x)\;:=\;a_{0}x^{n}\!+\!a_{1}x^{n-1}\!+\!\ldots\!+\!a_{n-1}x\!+\!a_{n}$

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  $N=2,\,3,\,4,\,\ldots$  there exists a finite number of equations (1) such that

 $n\!+\!a_{0}\!+\!|a_{1}|\!+\ldots+\!|a_{n}|\;=\;N$

(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

 $S_{2},\,S_{3},\,S_{4},\,\ldots$

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