root of unity

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Noun[edit]

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 th roots of unity are , where .
    • 2001, Jean-Pierre 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, 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 .
    • 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.

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