About the Countability of the Algebraic Numbers

(Polynomials with integer coefficients, Prime Numbers, Rational Numbers and Transcendent Numbers)






Jean-François COLONNA

www.lactamme.polytechnique.fr

jean-francois.colonna@polytechnique.edu
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France

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[en français/in french]


Abstract: How to "count" the polynomials with integer coefficients? A relationship between prime numbers and transcendent numbers.


Keywords: Polynomials, Polynômes, Prime Numbers, Nombres Premiers, Rational Numbers, Nombres Rationnels, Algebraic Numbers, Nombres Algébriques, Transcendent Numbers, Nombres Transcendants.



Let P(X) be a polynomial of the nth-degree with integer coefficients:
                    P(X) = An*Xn + An-1*Xn-1 + An-2*Xn-2 + (...) + A2*X2 + A1*X1 + A0*X0
                    Ai E Z \-/ i
                    An # 0


Then, using the n+1 polynomial coefficients, let's define the following unique rational number:
                    R = 2A0 * 3A1 * 5A2 * 7A3 (...)
where the numbers {2,3,5,7,...} are the n+1 first prime numbers. Then obviously:
                    R = a/b with a E N*,b E N*,HCF(a,b)=1
For example:
                                           ---------------------------------------
                                          |                                       |
                                          |       --------------------------      |
                                          |      |                          |     |
                                          |      |       -------------      |     |
                                          |      |      |             |     |     |
                    P(X) = X2 - X - 1 = + 1*X2 - 1*X1 - 1*X0 ==> R = 2-1 * 3-1 * 5+1 = 5/(2*3) = 5/6
(by the way, the positive root of the equation P(X)=0 defines the golden ratio).

The number R belongs therefore to the following set:
                    Q' = {a/b|a E N*,b E N*,HCF(a,b)=1}


Conversely, any number R in Q' defines an unique polynomial with integer coefficients.

For example:
                                             ---------------------------------------------------------------
                                            |                                                               |
                                            |     ---------------------------------------------------       |
                                            |    |                                                   |      |
                                            |    |     ----------------------------------------      |      |
                                            |    |    |                                        |     |      |
                                            |    |    |     ----------------------------       |     |      |
                                            |    |    |    |                            |      |     |      |
                                            |    |    |    |       --------------       |      |     |      |
                                            |    |    |    |      |              |      |      |     |      |
                    R = 22/7 = (2*11)/7 = 2+1 * 30 * 50 * 7-1 * 11+1 ==> P(X) = + 1*X4 - 1*X3 + 0*X2 + 0*X1 + 1*X0 = X4 - X3 + 1


This process defines a bijection between Q' and the set of the polynomials with integer coefficients. Q' being a subset of Q (the rational numbers) and Q being countable, then the set of the polynomials with integer coefficients is countable.

At last, the real roots of the polynomials with integer coefficients are defining the so-called algebraic numbers. Let's recall that a polynomial of the nth-degree has n complex roots and then at the most n real roots. Then the algebraic numbers are countable (a well known result, obtained here by means of the prime numbers and of the rational numbers).



Let's recall a consequence of this result: transcendent numbers are non countable and do exist. As a matter of fact:
RealNumbers = AlgebraicNumbers U TranscendentNumbers
AlgebraicNumbers /\ TranscendentNumbers = 0
Real Numbers are not countable and Algebraic Numbers are countable
then:

Transcendent Numbers are not countable and do exist






Annexe:

Here are the first positive Rational Numbers using the same order than the one used for the demonstration of their countability:


1/1: P(X) = 0 [*] 1/2: P(X) = -1 1/3: P(X) = -X 1/4: P(X) = -2 1/5: P(X) = -X2 1/6: P(X) = -X-1 1/7: P(X) = -X3 1/8: P(X) = -3
2/1: P(X) = +1 2/2=1/1 2/3: P(X) = -X+1 2/4=1/2 2/5: P(X) = -X2+1 2/6=1/3 2/7: P(X) = -X3+1
3/1: P(X) = +X 3/2: P(X) = +X-1 3/3=1/1 3/4: P(X) = +X-2 3/5: P(X) = -X2+X 3/6=1/2
4/1: P(X) = +2 4/2=2/1 4/3: P(X) = -X+2 4/4=1/1 4/5: P(X) = -X2+2
5/1: P(X) = +X2 5/2: P(X) = +X2-1 5/3: P(X) = +X2-X 5/4: P(X) = +X2-2
6/1: P(X) = +X+1 6/2=3/1 6/3=2/1
7/1: P(X) = +X3 7/2: P(X) = +X3-1
8/1: P(X) = +3


or again:


1/1: P(X) = 0 [*]
2/1: P(X) = +1 1/2: P(X) = -1
3/1: P(X) = +X 2/2=1/1 1/3: P(X) = -X
4/1: P(X) = +2 3/2: P(X) = +X-1 2/3: P(X) = -X+1 1/4: P(X) = -2
5/1: P(X) = +X2 4/2=2/1 3/3=1/1 2/4=1/2 1/5: P(X) = -X2
6/1: P(X) = +X+1 5/2: P(X) = +X2-1 4/3: P(X) = -X+2 3/4: P(X) = +X-2 2/5: P(X) = -X2+1 1/6: P(X) = -X-1
7/1: P(X) = +X3 6/2=3/1 5/3: P(X) = +X2-X 4/4=1/1 3/5: P(X) = -X2+X 2/6=1/3 1/7: P(X) = -X3
8/1: P(X) = +3 7/2: P(X) = +X3-1 6/3=2/1 5/4: P(X) = +X2-2 4/5: P(X) = -X2+2 3/6=1/2 2/7: P(X) = -X3+1 1/8: P(X) = -3


[*]: by convention


Copyright (c) Jean-François Colonna, 2020.
Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2020.