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
For example, when and , the cyclotomic field is a quadratic field
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 .
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
and 
, the cyclotomic field is a quadratic field
, the ring of integers (mod 
). Hence, a cyclotomic field is a Abelian extension. Not all cyclotomic fields have unique factorization, for instance,
where 
.