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For every positive integer n, there are n different n-th roots of unity. For example, the third roots of unity are 1, (-1 +i√3) /2 and (-1 - i√3) /2. In general, the n-th roots of unity can be written as:
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 n-th 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 n-th roots of unity, where φ(n) denotes Euler's phi function.
The n-th roots of unity are precisely the zeros of the polynomial p(X) = Xn - 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
By adjoining a primitive n-th root of unity to Q, one obtains the n-th cyclotomic field Fn. 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 Fn/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 Fn/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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