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
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
.