Editing Root of unity (section)

==General definition==
[[File:visualisation_complex_number_roots.svg|thumb|Geometric representation of the 2nd to 6th root of a general complex number in polar form. For the ''n''th root of unity, set {{mvar|r}}&nbsp;=&nbsp;1 and {{mvar|&phi;}}&nbsp;=&nbsp;0. The principal root is in black.]]

An ''{{mvar|n}}th root of unity'', where {{mvar|n}} is a positive integer, is a number {{mvar|z}} satisfying the [[equation]]<ref>{{Cite book|author=Hadlock, Charles R.|author-link=Charles Robert Hadlock|title=Field Theory and Its Classical Problems, Volume 14|publisher=Cambridge University Press|year=2000|isbn=978-0-88385-032-9|pages=84–86|url=https://books.google.com/books?id=5s1p0CyafnEC&pg=PA84}}</ref><ref>{{cite book|last = Lang|first = Serge|chapter=Roots of unity|title=Algebra|publisher=Springer|year=2002|isbn=978-0-387-95385-4|pages=276–277|chapter-url=https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA276}}</ref>
<math display="block">z^n = 1. </math>
Unless otherwise specified, the roots of unity may be taken to be [[complex number]]s (including the number 1, and the number −1 if {{mvar|n}} is [[parity (mathematics)|even]], which are complex with a zero [[complex number|imaginary part]]), and in this case, the {{mvar|n}}th roots of unity are<ref name="meserve">{{cite book
|last = Meserve |first = Bruce E.
|title = Fundamental Concepts of Algebra
|page = 52
|publisher = Dover Publications
|year = 1982}}</ref>
<math display="block">\exp\left(\frac{2k\pi i}{n}\right)=\cos\frac{2k\pi}{n}+i\sin\frac{2k\pi}{n},\qquad k=0,1,\dots, n-1.</math>

However, the defining equation of roots of unity is meaningful over any [[field (mathematics)|field]] (and even over any [[ring (mathematics)|ring]])  {{math|''F''}}, and this allows considering roots of unity in {{math|''F''}}. Whichever is the field {{math|''F''}}, the roots of unity in {{math|''F''}} are either complex numbers, if the [[characteristic (algebra)|characteristic]] of {{math|''F''}} is 0, or, otherwise, belong to a [[finite field]]. Conversely, every nonzero element in a finite field is a root of unity in that field. See [[Root of unity modulo n|Root of unity modulo ''n'']] and [[Finite field]] for further details.

An {{mvar|n}}th root of unity is said to be '''{{visible anchor|primitive}}''' if it is not an {{mvar|m}}th root of unity for some smaller {{mvar|m}}, that is if<ref name="moskowitz">{{cite book
|last = Moskowitz |first= Martin A.
|year = 2003
|title = Adventure in Mathematics
|publisher = World Scientific
|url = https://books.google.com/books?id=YT2_Kqsnn9wC&pg=PA36
|page = 36|isbn= 9789812794949
}}</ref><ref name="lidl">{{cite book
|last1 = Lidl |first1 = Rudolf
|last2 = Pilz |first2 = Günter |author-link2 = Günter Pilz
|year = 1984
|title = Applied Abstract Algebra
|series = Undergraduate Texts in Mathematics
|url = https://books.google.com/books?id=irXSBwAAQBAJ&pg=PA149
|page = 149
|publisher = Springer
|doi = 10.1007/978-1-4615-6465-2
|isbn = 978-0-387-96166-8
}}</ref>

:<math>z^n=1\quad \text{and} \quad z^m \ne 1 \text{ for } m = 1, 2, 3, \ldots, n-1. </math>
If ''n'' is a [[prime number]], then all {{math|''n''}}th roots of unity, except 1, are primitive.<ref name="morandi">{{cite book
|last = Morandi | first = Patrick
|title = Field and Galois theory
| series = Graduate Texts in Mathematics
|year = 1996
| volume = 167
|url = https://books.google.com/books?id=jQ7c8Xqpqk0C&pg=PA74
|page = 74
|publisher = Springer
|isbn = 978-0-387-94753-2
|doi = 10.1007/978-1-4612-4040-2}}</ref>

In the above formula in terms of exponential and trigonometric functions, the primitive {{mvar|n}}th roots of unity are those for which {{mvar|k}} and {{mvar|n}} are [[coprime integers]].

Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see {{slink|Finite field|Roots of unity}}. For the case of roots of unity in rings of [[modular arithmetic|modular integers]], see [[Root of unity modulo n|Root of unity modulo ''n'']].