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What is the formula to find a specific root of unity? Also, what does a primitive root of unity mean?

I know that $\zeta_5^5=1$ (5th root of unity), but how would I find $\zeta_5^2$? (the second 5th-root of unity)?

I'm just trying to grasp the concept. Any help would be appreciated. Thank you in advance!

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Have you tried to use the polar representation, $re^{i\theta},$ for a complex number? Try thinking about what $1$ is in polar form. –  Mike B Apr 24 '12 at 22:40
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Did you read Wikipedia? –  lhf Apr 24 '12 at 22:45
    
Yes...it didn't really give an explicit formula –  Dylan Apr 24 '12 at 22:47
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@dylan, it does if you care to read all lot it. –  lhf Apr 24 '12 at 23:40
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1 Answer

All $m$-th roots of unit are given by $\exp{\frac{2k\pi i}{m}}$ for $k=0,\dots m-1$. They correspond to the vertices of a regular $m$-gon inscribed in the unit circle and having $1$ as a vertex.

They are all powers of $\zeta_m = \exp{\frac{2\pi i}{m}}$, which is a primitive $m$-th root of unit. The other primitive $m$-th roots of unit are obtained by taking $k$ coprime with $m$.

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I could not edit because I only added 2 characters, but I think you meant $\exp{\frac{2k\pi i}{m}}$ instead of $\exp{\frac{2k\pi}{m}}$ and likewise for the other $\exp$. –  Tpofofn Apr 25 '12 at 0:53
    
@Tpofofn, fixed, thanks. –  lhf Apr 25 '12 at 1:04
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