# Abelian extension

﻿
Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable.

Any finite extension of a finite field is a cyclic extension. The development of class field theory has provided detailed information about abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields.

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 fields 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). 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 gives a complete description of the abelian extension case, and the Kronecker–Weber theorem tells us that if K is the field of rational numbers, 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 topology, which classifies all covering spaces of a space: abelian covers are classified by its abelianisation which relates directly to the first homology group.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Abelian — Abelian, in mathematics, is used in many different definitions, named after Norwegian mathematician Niels Henrik Abel:In group theory:*Abelian group, a group in which the binary operation is commutative **Category of abelian groups Ab has abelian …   Wikipedia

• Extension (mathematics) — In mathematics, the word extension has many uses. See:Analysis* Carathéodory s extension theorem * Continuous linear extension * M. Riesz extension theorem * Krein extension theorem * Hahn Banach theoremAlgebra* Abelian extension * Algebraic… …   Wikipedia

• Abelian group — For other uses, see Abelian (disambiguation). Abelian group is also an archaic name for the symplectic group Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product,… …   Wikipedia

• Abelian variety — In mathematics, particularly in algebraic geometry, complex analysis and number theory, an Abelian variety is a projective algebraic variety that is at the same time an algebraic group, i.e., has a group law that can be defined by regular… …   Wikipedia

• Extension of scalars — In abstract algebra, extension of scalars is a means of producing a module over a ring S from a module over another ring R, given a homomorphism f : R o S between them. Intuitively, the new module admits multiplication by more scalars than the… …   Wikipedia

• Timeline of abelian varieties — This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.Early history* c. 1000 Al Karaji writes on congruent numbers [ [http://www.cms.math.ca/Events/summer05/abs/pdf/hm.pdf PDF] ] eventeenth… …   Wikipedia

• Non-abelian class field theory — In mathematics, non abelian class field theory is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field K, to the general Galois… …   Wikipedia

• Group extension — In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence:1 ightarrow N… …   Wikipedia

• Equations defining abelian varieties — In mathematics, the concept of abelian variety is the higher dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ge; …   Wikipedia

• Semistable abelian variety — In mathematics, a semistable abelian variety in Diophantine geometry is an abelian variety defined over a global or local field with reduction modulo all primes of restricted type.For an Abelian variety A defined over a field F with ring of… …   Wikipedia