# About the Countability of the Algebraic Numbers (Polynomials with integer coefficients, Prime Numbers, Rational Numbers and Transcendent Numbers)

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#### 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