# Elementary Question about Roots of unity

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
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
@dylan, it does if you care to read all lot it. – lhf Apr 24 '12 at 23:40

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 $m$.
 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