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Famous Theorems of Mathematics/π is transcendental/Elementary symmetric polynomials

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Let there be a polynomial of degree

By the Fundamental Theorem of Algebra, it has complex roots (with multiplicity). Then we can write:

As we know, Vieta's formulae link between the coefficients and roots of a polynomial:

As we can see, these sums are symmetric polynomial, and are called elementary symmetric polynomials.

Definition

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The elementary symmetric polynomials in variables , are defined as such:


Elementary symmetric polynomials