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 extension field 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,
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 
.
. Its Galois group has two elements, the nontrivial element sending 
to 
, and is Abelian. By contrast, the degree-six extension 
, 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.
, 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 
.
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,