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 we will call them elementary symmetric polynomials.
Definition
[edit | edit source]The elementary symmetric polynomials in variables , are defined as such:
←Symmetric polynomials | Elementary symmetric polynomials | Fundamental theorem of symmetric polynomials→ |