abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, an abelian extension is a

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...

whose

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal

extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (proof theory) * Extension (predicate logic), the set of tuples of values t ...

of an abelian group. Every finite extension of a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

is a cyclic extension.

Description

Class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

provides detailed information about the abelian extensions of

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

s, function fields of

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

s over finite fields, and

local field In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...

s. There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining

roots of unity In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group char ...

to a field, or a subextension of such an extension. The

cyclotomic field In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...

s are examples. A cyclotomic extension, under either definition, is always 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

Kummer extension Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873–1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Chris ...

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 In field theory (mathematics), field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial (field theory), minimal polynomial of \alpha over is a separable po ...

). 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 is a German surname. Notable people with the surname include: * Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873–1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Chri ...

gives a complete description of the abelian extension case, and the

Kronecker–Weber theorem In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form modular arithmetic, (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provide ...

tells us that if ''K'' is the field of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

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 In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, which classifies all covering spaces of a space: abelian covers are classified by its

abelianisation In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

which relates directly to the first

homology group In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...

References

* *{{MathWorld , id=AbelianExtension , title=Abelian Extension Field extensions Algebraic number theory Class field theory