If 
is an algebraic Galois
extension field of 
such that the Galois group
of the extension is Abelian, then 
is said to be an Abelian extension of 
.
For example,
 |
is the field of rational numbers with the square root of two adjoined, a degree-two extension of 
.
Its Galois group has two elements, the nontrivial
element sending 
to 
,
and is Abelian. By contrast, the degree-six extension
 |
is the splitting field of 
, and is not an Abelian extension of 
. Indeed, the six automorphisms of 
, fixing 
, are defined by the permutations of the three roots of 
. So the Galois
group in this case is the symmetric group
on three letters, which is non-Abelian.
In an Abelian extension that is a splitting field for a polynomial 
,
the roots of 
are related. For instance, consider a cyclotomic
field, 
,
where 
is a primitive root 
and 
is a prime number. Then the
Galois group is the multiplicative group of the cyclic
group 
.
A classical theorem in number theory says that an Abelian extension of the rationals must be a subfield of a cyclotomic
field. Abelian extensions are in a sense the simplest kind of extension because
Abelian groups are easier to understand than more general ones. One nice property
of an Abelian extension 
of a field 
is that any intermediate subfield 
, with 
, must be a Galois
extension field of 
and, by the fundamental
theorem of Galois theory, also an Abelian extension,
