The mathematical constant is a transcendental number (or inalgebraic).
In other words, it is not a root of any polynomial with rational coefficients.
Let us assume that is algebraic, so there exists a polynomial
such that .
Lemma: If is algebraic, then is algebraic.
Proof: We get
hence is a root of the polynomial
Therefore, there exists a polynomial of degree with roots , such that .
By Euler's identity we have . Therefore:
The exponents are symmetric polynomial in , and among them are non-zero sums. That is:
As we previously learned, for all there exists a monic polynomial of degree such that its roots are the sums of every of the roots . Therefore:
After reduction we get that:
Multiplying by the least common multiple of the rational coefficients, we get a polynomial of the form
Let be a polynomial of degree . Let us define . Taking its derivative yields:
Let us define . Taking its derivative yields:
By fundamental theorem of calculus, we get:
Now let:
Summing all the terms yields:
Lemma: Let be a polynomial with a root of multiplicity . Then for all .
Proof: By strong induction.
Let us write , with a polynomial such that .
For we get:
Assume that for all the claim holds for all .
We shall prove that for the claim holds for all :
The blue part is of multiplicity , with a polynomial such that .
Hence their product satisfies the induction hypothesis.
Let us now define:
and is a prime number such that . We get:
hence for all , the function is a polynomial with integer coefficients all divisible by .
By parts 1 and 3, we get:
The red part is an integer not divisible by .
The green part is an integer divisible by .
The blue part is the most important:
By Vieta's formulae we get
and the sums are symmetric polynomials in . Therefore, these can be expressed as polynomials
In addition, we get:
Hence, the blue part is an integer divisible by .
Conclusion: is an integer not divisible by , and particularly .
By part 2, we get:
By the triangle inequality for integrals, we get:
By the triangle inequality, we get:
On the other hand, we get
hence for sufficiently large we get . A contradiction.
Conclusion: is transcendental, hence is transcendental.