In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, an abelian extension is a Galois extension

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...

 whose Galois group

Galois group

In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

 is abelian

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

. When the Galois group is a cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...

.

Any finite extension of a finite field

Finite field

In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

 is a cyclic extension. The development of class field theory

Class field theory

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...

 has provided detailed information about abelian extensions of number fields, function fields

Function field of an algebraic variety

In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...

 of algebraic curve

Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

s over finite fields, and local field

Local field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

s.

There are two slightly different concepts of cyclotomic extensions: these can mean either extensions formed by adjoining roots of unity, or subextensions of such extensions. The cyclotomic field

Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...

s are examples. Any cyclotomic extension (for either definition) is abelian.

If a field K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resulting so-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n, since otherwise this can fail even to be a separable extension

Separable extension

In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...

). In general, however, the Galois groups of n-th roots of elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-direct product. The Kummer theory

Kummer theory

In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last...

 gives a complete description of the abelian extension case, and the Kronecker–Weber theorem

Kronecker–Weber theorem

In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...

 tells us that if K is the field of rational number

Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

s, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.

There is an important analogy with the fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

 in topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, which classifies all covering spaces of a space: abelian covers are classified by its abelianisation which relates directly to the first homology group.