root of unity

English

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 $n$ $n$th roots of unity are $\textstyle \cos \left(k{\frac {2\pi }{n}}\right)+i\sin \left(k{\frac {2\pi }{n}}\right)$, where $k=1,\dots ,n$.

   • 2001, Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, page 89:

     We now show that the primitive $n$-th roots of unity generate the other $n$-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 $\mathbb {Q} _{p}$.

   • 2007, Carl L. DeVito, Harmonic Analysis: A Gentle Introduction, Jones & Bartlett Learning, page 150:

     We have seen that, for a fixed value of $n$, the multiplicative group $(U_{n},{\dot {)}}$ is generated by any primitive nth root of unity. In particular, if $\omega$ is a primitive 6th root of unity, then $\omega ^{6}=1$, six is the smallest positive integer for which this is true, and $U_{6}=\{\omega ^{0},\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5}\}$. It is easy to see that $\omega ^{2}$, which is a 6th root of unity, is also a cube root of unity. The same is true of $\omega ^{4}$. The element $\omega ^{3}$ is a square root of unity, whereas $\omega ^{5}$ is primitive.

Hypernyms

• algebraic integer

Holonyms

• circle group
• Kummer ring

Translations

field element, some positive power of which equals 1

• French: racine de l’unité f
• German: Einheitswurzel f
• Italian: radice dell'unità f
• Spanish: raíz de la unidad f

Further reading

• Argand system on Wikipedia.Wikipedia
• Circle group on Wikipedia.Wikipedia
• Multiplicative group on Wikipedia.Wikipedia