# 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 ${\displaystyle n}$ ${\displaystyle n}$th roots of unity are ${\displaystyle \textstyle \cos \left(k{\frac {2\pi }{n}}\right)+i\sin \left(k{\frac {2\pi }{n}}\right)}$, where ${\displaystyle k=1,\dots ,n}$.
• 2001, Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, page 89,
We now show that the primitive ${\displaystyle n}$-th roots of unity generate the other ${\displaystyle n}$-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 ${\displaystyle \mathbb {Q} _{p}}$.
• 2007, Carl L. DeVito, Harmonic Analysis: A Gentle Introduction, Jones & Bartlett Learning, page 150,
We have seen that, for a fixed value of ${\displaystyle n}$, the multiplicative group ${\displaystyle (U_{n},{\dot {)}}}$ is generated by any primitive nth root of unity. In particular, if ${\displaystyle \omega }$ is a primitive 6th root of unity, then ${\displaystyle \omega ^{6}=1}$, six is the smallest positive integer for which this is true, and ${\displaystyle U_{6}=\{\omega ^{0},\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5}\}}$. It is easy to see that ${\displaystyle \omega ^{2}}$, which is a 6th root of unity, is also a cube root of unity. The same is true of ${\displaystyle \omega ^{4}}$. The element ${\displaystyle \omega ^{3}}$ is a square root of unity, whereas ${\displaystyle \omega ^{5}}$ is primitive.