Complex Numbers

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  • #70889
    Shreya Nair
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    #74433
    Shirsendu Roy
    Moderator

    The product of norms-norm of product

    \prod_{k=1}^{n-1}||e^\frac}2\piik}{n}-1||=||\prod_{k=1}^{n-1}(e^\frac{2\piik}{n}-1)||

    Let w_k=e^\frac{2\piik}{n} for roots of unity other than 1, these are the roots of the equation z^{n-1}+z^{n-2}+…+z+1=0

    so,(w_k-1) is a root of

    (z+1)^{n-1}+(z+1)^{n-2}+…+(z+1)+1=0 for all k.

    The product of the roots of the polynomial are by Vieta’s formula equal to (-1)^{n-1}\frac{a_0}{a_n} where a_0 is the coefficient on the constant term a_n is the coefficient on the leading term

    leading term’s coefficient is 1, the constant term is the sum of n 1’s.

    product is (-1)^{n-1}n and its norm is n.

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