An Abelian extension of a field 
is an algebraic Galois extension field 
whose Galois group is Abelian.
For example,
 |
is the field of rational numbers with the square root of two adjoined,
an extension field of extension
field degree 2 of 
.
Its Galois group has two elements,
the nontrivial element sending 
to 
, and is Abelian. By
contrast, the extension field of extension
field degree 6
 |
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 polynomial
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 polynomial roots
of 
are related. For instance, consider a cyclotomic
field, 
,
where 
is a primitive root of unity 
and 
is a prime number. Then the
Galois group is the multiplicative
group of units of the quotient
ring 
.
This group is a cyclic group
of group order 
.
The Kronecker-Weber theorem says that every Abelian finite extension of 
is 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.
