# root of unity

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### Noun

root of unity (plural roots of unity)

1. (number theory) An element of a given field (especially, a complex number) x such that for some positive integer n, xn = 1.
In the case of the field of complex numbers, it follows from de Moivre's formula that the $n$ $n$ th roots of unity are $\textstyle \cos \left(k{\frac {2\pi }{n}}\right)+i\sin \left(k{\frac {2\pi }{n}}\right)$ , where $k=1,\dots ,n$ .
• 2001, Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, page 89,
We now show that the primitive $n$ -th roots of unity generate the other $n$ -th roots of unity.
• 2003, Fernando Gouvêa, p-adic Numbers: An Introduction, Springer, page 72,
A nice application of Hensel's Lemma is to determine which roots of unity can be found in $\mathbb {Q} _{p}$ .
• 2007, Carl L. DeVito, Harmonic Analysis: A Gentle Introduction, Jones & Bartlett Learning, page 150,
We have seen that, for a fixed value of $n$ , the multiplicative group $(U_{n},{\dot {)}}$ is generated by any primitive nth root of unity. In particular, if $\omega$ is a primitive 6th root of unity, then $\omega ^{6}=1$ , six is the smallest positive integer for which this is true, and $U_{6}=\{\omega ^{0},\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5}\}$ . It is easy to see that $\omega ^{2}$ , which is a 6th root of unity, is also a cube root of unity. The same is true of $\omega ^{4}$ . The element $\omega ^{3}$ is a square root of unity, whereas $\omega ^{5}$ is primitive.