A cyclotomic field 
is obtained by adjoining a primitive root of
unity 
of order 
to the rational numbers 
. It contains all the 
th roots of unity, since these
are powers of 
,
 |
(1)
|
For example, when 
and 
, the cyclotomic field is a quadratic
field
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
The Galois group of a cyclotomic field over the rational numbers is the multiplicative
group of units of the quotient
ring 
. Hence a cyclotomic field is an Abelian
extension. Its ring of integers need not have
unique factorization. For example, the ring of integers of 
fails to have unique
factorization when 
is a primitive 23rd root of
unity (Lemmermeyer 1998).
