root of unity
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English[edit]
Noun[edit]
root of unity (plural roots of unity)
 (number theory) An element of a given field (especially, a complex number) x such that for some positive integer n, x^{n} = 1.
 In the case of the field of complex numbers, it follows from de Moivre's formula that the th roots of unity are , where .
 2001, JeanPierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, page 89,
 We now show that the primitive th roots of unity generate the other th roots of unity.
 2003, Fernando Gouvêa, padic Numbers: An Introduction, Springer, page 72,
 A nice application of Hensel's Lemma is to determine which roots of unity can be found in .
 2007, Carl L. DeVito, Harmonic Analysis: A Gentle Introduction, Jones & Bartlett Learning, page 150,
 We have seen that, for a fixed value of , the multiplicative group is generated by any primitive nth root of unity. In particular, if is a primitive 6th root of unity, then , six is the smallest positive integer for which this is true, and . It is easy to see that , which is a 6th root of unity, is also a cube root of unity. The same is true of . The element is a square root of unity, whereas is primitive.
Hypernyms[edit]
Holonyms[edit]
Translations[edit]
field element, some positive power of which equals 1


Further reading[edit]
 Argand system on Wikipedia.Wikipedia
 Circle group on Wikipedia.Wikipedia
 Multiplicative group#Group scheme of roots of unity on Wikipedia.Wikipedia
 Kummer ring on Wikipedia.Wikipedia