root of unity

Definitions

An element of a given field (especially, a complex number) x such that for some positive integer n, xⁿ = 1.

Equivalents

Examples

“In the case of the field of complex numbers, it follows from de Moivre's formula that the n nth roots of unity are #92;textstyle#92;cos#92;left(k#92;frac#123;2#92;pi#125;#123;n#125;#92;right)#43;i#92;sin#92;left(k#92;frac#123;2#92;pi#125;#123;n#125;#92;right), where k#61;1,#92;dots,n.”

“We now show⟳ that the primitive n-th roots of unity generate⟳ the other n-th roots of unity.”

“A nice application of Hensel's Lemma is to determine⟳ which roots of unity can be found in #92;Q#95;p.”

“We have⟳ seen that, for a fixed value of n, the multiplicative group (U#95;n,#92;dot) is generated by any primitive nth root of unity. In particular, if #92;omega is a primitive 6th root of unity, then #92;omega⁶#61;1, six is the smallest positive integer for which this is true, and U#95;6#61;#92;#123;#92;omega⁰,#92;omega,#92;omega²,#92;omega³,#92;omega⁴,#92;omega⁵#92;#125;. It is easy to see⟳ that #92;omega², which is a 6th root of unity, is also a cube root of unity. The same is true of #92;omega⁴. The element #92;omega³ is a square root of unity, whereas #92;omega⁵ is primitive.”

CEFR level