A cyclotomic field is obtained by adjoining a primitive root of unity , say , to the rational numbers . Since is primitive, is also an nth root of unity and contains all of the nth roots of unity,

For example, when n = 3 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 n). Hence, a cyclotomic field is a Abelian extension. Not all cyclotomic fields have unique factorization, for instance, where .

 

Extension Field, Number Field