Table of contents |

2 Algebraic structure 3 Cyclotomic polynomials 4 Cyclotomic fields |

Providing *n* is at least 2, these numbers add up to 0, a simple fact that is of constant use in mathematics. It can be proved in any number of ways, for example by recognising the sum as coming from a geometric progression.

The *n*th roots of unity form a group under multiplication of complex numbers. This group is cyclic. A generator of this group is called a **primitive n-th root of unity**. The primitive

The *n*-th roots of unity are precisely the zeros of the polynomial *p*(*X*) = *X*^{n} − 1; the primitive *n*-th roots of unity are precisely the zeros of the *n*th cyclotomic polynomial

Every *n*th root of unity is a primitive *d*th root of unity for exactly one positive divisor *d* of *n*. This implies that

- Φ
_{1}(*X*) =*X*− 1 - Φ
_{2}(*X*) =*X*+ 1 - Φ
_{3}(*X*) =*X*^{2}+*X*+ 1 - Φ
_{4}(*X*) =*X*^{2}+ 1 - Φ
_{5}(*X*) =*X*^{4}+*X*^{3}+*X*^{2}+*X*+ 1 - Φ
_{6}(*X*) =*X*^{2}-*X*+ 1

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

As the Galois group of *F*_{n}/**Q** is abelian, this is an abelian extension. Every subfield of a cyclotomic field is an abelian extension of the rationals. In these cases Galois theory can be written out quite 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 - a theorem of Kronecker.