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Mathematics-Online lexicon:

Complex Roots of Unity

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The equation

$\displaystyle z^n = 1 $

has exactly $ n$ solutions in $ \mathbb{C}$

$\displaystyle z_k = w_n^k,\quad w_n = \exp(2\pi\mathrm{i}/n), \quad k=0,\ldots,n-1\,. $

Those are called complex roots of unity

\includegraphics[width=0.5\linewidth]{Bild_Einheitswurzel.eps}

As illustrated in the figure the $ n$ -th roots of unity form a regular $ n$ -polygon inscribed in the unit circle.

(Authors: Höllig/Kopf/Abele)
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  automatically generated 10/ 4/2007