Editing Root of unity (section)

==Cyclotomic fields==
{{Main|Cyclotomic field}}
By [[Field_extension#adjunction|adjoining]] a primitive {{mvar|n}}th root of unity to <math>\Q,</math> one obtains the {{mvar|n}}th [[cyclotomic field]] <math>\Q(\exp(2\pi i/n)).</math>This [[field (mathematics)|field]] contains all {{mvar|n}}th roots of unity and is the [[splitting field]] of the {{mvar|n}}th cyclotomic polynomial over <math>\Q.</math> The [[field extension]] <math>\Q(\exp(2\pi i /n))/\Q</math> has degree φ(''n'') and its [[Galois group]] is [[Natural transformation|naturally]] [[group isomorphism|isomorphic]] to the multiplicative [[group of units]] of the ring <math>\Z/n\Z.</math>

As the Galois group of <math>\Q(\exp(2\pi i /n))/\Q</math> is abelian, this is an [[abelian extension]]. Every [[field extension|subfield]] of a cyclotomic field is an abelian extension of the rationals. It follows that every ''n''th root of unity may be expressed in term of ''k''-roots, with various ''k'' not exceeding φ(''n''). In these cases [[Galois theory]] can be written out explicitly in terms of [[Gaussian period]]s: this theory from the ''[[Disquisitiones Arithmeticae]]'' of [[Carl Friedrich Gauss|Gauss]] was published many years before Galois.<ref>The ''Disquisitiones'' was published in 1801, [[Évariste Galois|Galois]] was born in 1811, died in 1832, but wasn't published until 1846.</ref>

Conversely, ''every'' abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of [[Leopold Kronecker|Kronecker]], usually called the ''[[Kronecker–Weber theorem]]'' on the grounds that Weber completed the proof.