Editing Root of unity (section)

==Group properties==
===Group of all roots of unity===
The product and the [[multiplicative inverse]] of two roots of unity are also roots of unity. In fact, if {{math|''x<sup>m</sup>'' {{=}} 1}} and {{math|''y<sup>n</sup>'' {{=}} 1}}, then {{math|(''x''<sup>−1</sup>){{sup|''m''}} {{=}} 1}}, and {{math|(''xy''){{sup|''k''}} {{=}} 1}}, where {{math|''k''}} is the [[least common multiple]] of {{math|''m''}} and {{math|''n''}}.

Therefore, the roots of unity form an [[abelian group]] under multiplication. This [[group (mathematics)|group]] is the [[torsion subgroup]] of the [[circle group]].

===Group of {{math|''n''}}th roots of unity===
For an integer ''n'', the product and the multiplicative inverse of two {{math|''n''}}th roots of unity are also {{math|''n''}}th roots of unity. Therefore, the {{math|''n''}}th roots of unity form an abelian group under multiplication.

Given a primitive {{math|''n''}}th root of unity {{math|''ω''}}, the other {{math|''n''}}th roots are powers of {{math|''ω''}}. This means that the group of the {{math|''n''}}th roots of unity is a [[cyclic group]]. It is worth remarking that the term of ''cyclic group'' originated from the fact that this group is a [[subgroup]] of the [[circle group]].

===Galois group of the primitive {{math|''n''}}th roots of unity===
Let <math>\Q(\omega)</math> be the [[field extension]] of the [[rational number]]s generated over <math>\Q</math> by a primitive {{math|''n''}}th root of unity {{math|''ω''}}. As every {{math|''n''}}th root of unity is a power of {{math|''ω''}}, the [[field (mathematics)|field]] <math>\Q(\omega)</math> contains all {{math|''n''}}th roots of unity, and <math>\Q(\omega)</math> is a [[Galois extension]] of <math>\Q.</math>

If {{math|''k''}} is an integer, {{math|''ω<sup>k</sup>''}} is a primitive {{math|''n''}}th root of unity if and only if {{math|''k''}} and {{math|''n''}} are [[coprime]]. In this case, the map

:<math>\omega \mapsto \omega^k</math>

induces an [[field automorphism|automorphism]] of <math>\Q(\omega)</math>, which maps every {{math|''n''}}th root of unity to its {{math|''k''}}th power. Every automorphism of <math>\Q(\omega)</math> is obtained in this way, and these automorphisms form the [[Galois group]] of <math>\Q(\omega)</math> over the field of the rationals.

The rules of exponentiation imply that the [[function composition|composition]] of two such automorphisms is obtained by multiplying the exponents. It follows that the map

:<math>k\mapsto \left(\omega \mapsto \omega^k\right)</math>

defines a [[group isomorphism]] between the [[unit (ring theory)|units]] of the ring of [[integers modulo n|integers modulo {{math|''n''}}]] and the Galois group of <math>\Q(\omega).</math>

This shows that this Galois group is [[abelian group|abelian]], and implies thus that the primitive roots of unity may be expressed in terms of [[radical expression|radicals]].

===Galois group of the real part of the primitive roots of unity===
{{main|Minimal polynomial of 2cos(2pi/n)}}
The real part of the primitive roots of unity are related to one another as roots of the [[minimal polynomial (field theory)|minimal polynomial]] of <math>2\cos(2\pi/n).</math> The roots of the minimal polynomial are just twice the real part; these roots form a cyclic Galois group.