A cyclotomic field
is obtained by adjoining a primitive root of
unity ,
say ,
to the rational numbers . Since is primitive, is also an th root of unity and contains all of the th roots of unity,
(1)
For example, when
and ,
the cyclotomic field is a quadratic field
(2)
(3)
(4)
where the coefficients are contained in .
The Galois group of a cyclotomic field over the rationals is the multiplicative group of ,
the ring of integers (mod ). Hence, a cyclotomic field is a Abelian
extension. Not all cyclotomic fields have unique factorization, for instance,
, where .