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Prove-that-the-product-for-all-nth-roots-of-unity-is-equal-to-zero-except-n-1-Note-U-n-e-2kpii-n-k-1-2-n-x-n-1-




Question Number 21756 by FilupS last updated on 03/Oct/17
Prove that the product for all  nth roots of unity is equal to zero,  except n=1.     Note:  U_n ={e^(2kπi/n)  ∣ k∈{1, 2, ..., n}}  x^n =1
$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{for}\:\mathrm{all} \\ $$$${n}\mathrm{th}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{zero}, \\ $$$$\mathrm{except}\:{n}=\mathrm{1}. \\ $$$$\: \\ $$$$\mathrm{Note}: \\ $$$${U}_{{n}} =\left\{{e}^{\mathrm{2}{k}\pi{i}/{n}} \:\mid\:{k}\in\left\{\mathrm{1},\:\mathrm{2},\:…,\:{n}\right\}\right\} \\ $$$${x}^{{n}} =\mathrm{1} \\ $$

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