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Cyclotomic fields

By adjoining a primitive n th root of unity to Q, {\displaystyle \mathbb {Q},} one obtains the n th cyclotomic field Q (exp (2 π i / n)). {\displaystyle \mathbb {Q} (\exp(2\pi i/n)).} This field contains all n th roots of unity and is the splitting field of the n th cyclotomic polynomial over Q. {\displaystyle \mathbb {Q}.} The field extension Q (exp (2 π i / n)) / Q {\displaystyle \mathbb {Q} (\exp(2\pi i/n))/\mathbb {Q} } has degree φ(n) and its Galois group is naturally isomorphic to the multiplicative group of units of the ring Z / n Z. {\displaystyle \mathbb {Z} /n\mathbb {Z}.}

As the Galois group of Q (exp (2 π i / n)) / Q {\displaystyle \mathbb {Q} (\exp(2\pi i/n))/\mathbb {Q} } is abelian, this is an abelian extension. Every 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 periods: this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.

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