For every positive integer n, there are n different nth roots of unity. For example, the third roots of unity are 1, (1 +i√3) /2 and (1  i√3) /2. In general, the nth roots of unity can be written as:
The nth roots of unity form a group under multiplication of complex numbers. This group is cyclic. A generator of this group is called a primitive nth root of unity. The primitive nth roots of unity are precisely the numbers of the form exp(2πij/n) where j and n are coprime. Therefore, there are φ(n) different primitive nth roots of unity, where φ(n) denotes Euler's phi function.
The nth roots of unity are precisely the zeros of the polynomial p(X) = X^{n}  1; the primitive nth roots of unity are precisely the zeros of the nth cyclotomic polynomial
Every nth root of unity is a primitive dth root of unity for exactly one positive divisor d of n. This implies that
By adjoining a primitive nth root of unity to Q, one obtains the nth cyclotomic field F_{n}. This field contains all nth roots of unity and is the splitting field of the nth 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/nZ.
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[?].
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