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An $n$-th root of unity is a complex number $z$ such that $z^n=1$ for some $n\in \mathbb N$. If $n$ cannot be replaced by a smaller natural number, then $z$ is called primitive $n$-th root of unity. There are $\varphi(n)$ primitive $n$-th roots of unity and they are roots of the $n$-th cyclotomic polynomial (which has degree $\varphi(n)$). The $n$-th roots of unity can be written as $e^{ \frac{2k\pi}n\cdot i}$ with $0\le k\lt n$.

The concept can be extended to other fields than $\mathbb C$. For example, in a finite field with $q$ elements, all non-zero elements are $(q-1)$-th roots of unity.

See also http://en.wikipedia.org/wiki/Root_of_unity

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