Editing Root of unity (section)

===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]].