Root of unity: Difference between revisions
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In [[mathematics]], a '''root of unity''' is an algebraic quantity some power of which is equal to one. An ''n''-th root of unity is a number ζ such that ζ<sup>''n''</sup> = 1. A ''primitive'' ''n''-th root of unity is one which is an ''n''-th root but not an ''m''-th root for any ''m'' less than ''n''. Any ''n''-th root of unity is a primitive ''d''-th root of unity for some ''d'' dividing ''n''. | In [[mathematics]], a '''root of unity''' is an algebraic quantity some power of which is equal to one. An ''n''-th root of unity is a number ζ such that ζ<sup>''n''</sup> = 1. A ''primitive'' ''n''-th root of unity is one which is an ''n''-th root but not an ''m''-th root for any ''m'' less than ''n''. Any ''n''-th root of unity is a primitive ''d''-th root of unity for some ''d'' dividing ''n''. | ||
The ''n''-th roots of unity are the roots of the [[polynomial]] ''X''<sup>''n''</sup> - 1; the primitive ''n''-th roots of unity are the roots of the [[cyclotomic polynomial]] Φ<sub>''n''</sub>(''X''). | |||
Roots of unity are clearly [[algebraic number]]s, and indeed [[algebraic integer]]s. It is often convenient to identify the ''n''-th roots of unity with the [[complex number]]s exp(2πi.''r''/''n'') with ''r''=0,...,''n''-1 and the primitive ''n''-th roots with those numbers of the form exp(2πi.''r''/''n'') with ''r'' [[coprime]] to ''n''. However, the concept of root of unity makes sense in other context such as [[p-adic field]]s and [[finite field]]s (in the latter case every non-zero element is a root of unity). | |||
Revision as of 13:53, 11 December 2008
In mathematics, a root of unity is an algebraic quantity some power of which is equal to one. An n-th root of unity is a number ζ such that ζn = 1. A primitive n-th root of unity is one which is an n-th root but not an m-th root for any m less than n. Any n-th root of unity is a primitive d-th root of unity for some d dividing n.
The n-th roots of unity are the roots of the polynomial Xn - 1; the primitive n-th roots of unity are the roots of the cyclotomic polynomial Φn(X).
Roots of unity are clearly algebraic numbers, and indeed algebraic integers. It is often convenient to identify the n-th roots of unity with the complex numbers exp(2πi.r/n) with r=0,...,n-1 and the primitive n-th roots with those numbers of the form exp(2πi.r/n) with r coprime to n. However, the concept of root of unity makes sense in other context such as p-adic fields and finite fields (in the latter case every non-zero element is a root of unity).