root of unity

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

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

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

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

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