Hi, can someone please help me with the following question?(adsbygoogle = window.adsbygoogle || []).push({});

Q. Let [itex]\omega _0 ,...,\omega _{n - 1} [/itex] be the nth roots of 1. Show that

[tex]

\sum\limits_{j = 0}^{n - 1} {\omega _j ^k } = \left\{ {\begin{array}{*{20}c}

{0,1 \le k \le n - 1} \\

{n,k = n} \\

\end{array}} \right.

[/tex]

The case of k = n is fairly easy but I don't know where to start when I attempt the case of 1 <= k <= n-1. For k = 1 I can use a previous result but for 'most' cases I can't. Any help would be good thanks.

Edit: For k =1, the result follows from a previous part of the question and k = n is an easy case. For 2 <= k <= n-1 I just said [itex]\omega _j ^k \ne \omega _j[/itex] but (w_j)^k must be equal to one of the other roots. For clarity, I considered (w_0)^k != w_0 for 2 <= k <= n-1 and then said (w_0)^k must be equal to one of the other roots. Applying the same reasoning to the other w_j and using that the sum of the roots is equal to zero leads to the given result.

That's about all I've been able to come up with. That's not correct because one of the nth roots of 1 is 1 and 1^2 = 1. I just thought that I'd put something up so that someone will help me out.

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# Homework Help: Working with Complex numbers

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