A cyclotomic field Q(ζ) is obtained by adjoining a primitive root of unity ζ, say ζ^n = 1, to the rational numbers Q. Since ζ is primitive, ζ^k is also an nth root of unity and Q(ζ) contains all of the nth roots of unity, Q(ζ) = { sum_(k = 0)^(n - 1) a_i ζ^k :a_i element Q}. For example, when n = 3 and ζ = (-1 + isqrt(3))/2, the cyclotomic field is a quadratic field Q(ζ) | = | {a_0 + a_1 ζ + a_2 ζ^2} | = | {b_0 + b_1 sqrt(-3)} | = | Q(sqrt(-3)), where the coefficients b_i are contained in Q.

extension field | number field