Famous Theorems of Mathematics/π is transcendental/Proof

The mathematical constant is a transcendental number (or inalgebraic). In other words, it is not a root of any polynomial with rational coefficients. Proof Let us assume that is algebraic, so there exists a polynomial such that . Part 1 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 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 Part 2 Let be a polynomial of degree . Let us define: Taking its derivative yields: Let us define . Taking its derivative yields: By the Fundamental Theorem of Calculus, we get: Now let: Summing all the terms yields: Part 3 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 blue part is of multiplicity , with a polynomial such that . Hence their product satisfies the induction hypothesis. Part 4 Let us now define: such that is a prime number. We get: Notes: The coefficients of are integers all divisible by . For , the coefficients of are integers all divisible by and . By parts 1 and 3, we get: Now let us choose a prime number . We get: The red red part is an integer not divisible by . The green green part is an integer divisible by . The blue blue part is the most important: By Vieta's formulae we get and the blue sums are symmetric polynomials in . By the Fundamental Theorem of Symmetric Polynomials, these can be expressed as polynomials such that: Therefore divides , and the DeepSkyBlue DeepSkyBlue part is an integer. Hence, the blue part is also an integer divisible by . Conclusion: is an integer not divisible by , and particularly . Hence . Part 5 In part 2, the integral is independent of the path between the points. Without loss of generality, let us choose the line interval . In the interval we get: Let us define: 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. Proof Fundamental theorem of symmetric polynomials