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• Abelian_extension abstract "In abstract algebra, an abelian extension is a Galois extension whose Galois group 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 of an abelian group.Every finite extension of a finite field is a cyclic extension.Class field theory provides detailed information about the abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields.There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields 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 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.".
• Abelian_extension wikiPageID "174448".
• Abelian_extension wikiPageLength "2385".
• Abelian_extension wikiPageOutDegree "28".
• Abelian_extension wikiPageRevisionID "708273904".
• Abelian_extension wikiPageWikiLink Abelian_group.
• Abelian_extension wikiPageWikiLink Abstract_algebra.
• Abelian_extension wikiPageWikiLink Algebraic_curve.
• Abelian_extension wikiPageWikiLink Algebraic_number_field.
• Abelian_extension wikiPageWikiLink Category:Algebraic_number_theory.
• Abelian_extension wikiPageWikiLink Category:Class_field_theory.
• Abelian_extension wikiPageWikiLink Category:Field_extensions.
• Abelian_extension wikiPageWikiLink Class_field_theory.
• Abelian_extension wikiPageWikiLink Commutator_subgroup.
• Abelian_extension wikiPageWikiLink Cyclic_group.
• Abelian_extension wikiPageWikiLink Cyclotomic_field.
• Abelian_extension wikiPageWikiLink Finite_field.
• Abelian_extension wikiPageWikiLink Function_field_of_an_algebraic_variety.
• Abelian_extension wikiPageWikiLink Fundamental_group.
• Abelian_extension wikiPageWikiLink Galois_extension.
• Abelian_extension wikiPageWikiLink Galois_group.
• Abelian_extension wikiPageWikiLink Group_extension.
• Abelian_extension wikiPageWikiLink Homology_(mathematics).
• Abelian_extension wikiPageWikiLink Kronecker–Weber_theorem.
• Abelian_extension wikiPageWikiLink Kummer_theory.
• Abelian_extension wikiPageWikiLink Local_field.
• Abelian_extension wikiPageWikiLink Rational_number.
• Abelian_extension wikiPageWikiLink Root_of_unity.
• Abelian_extension wikiPageWikiLink Semidirect_product.
• Abelian_extension wikiPageWikiLink Separable_extension.
• Abelian_extension wikiPageWikiLink Solvable_group.
• Abelian_extension wikiPageWikiLink Topology.
• Abelian_extension wikiPageWikiLinkText "Abelian extension".
• Abelian_extension wikiPageWikiLinkText "abelian extension".
• Abelian_extension wikiPageWikiLinkText "abelian".
• Abelian_extension wikiPageWikiLinkText "cyclic cyclotomic extension".
• Abelian_extension wikiPageWikiLinkText "cyclic extensions".
• Abelian_extension wikiPageWikiLinkText "cyclic".
• Abelian_extension wikiPageWikiLinkText "extensions".
• Abelian_extension first "L.V.".
• Abelian_extension id "C/c027560".
• Abelian_extension last "Kuz'min".
• Abelian_extension title "cyclotomic extension".
• Abelian_extension wikiPageUsesTemplate Template:Springer.
• Abelian_extension subject Category:Algebraic_number_theory.
• Abelian_extension subject Category:Class_field_theory.
• Abelian_extension subject Category:Field_extensions.
• Abelian_extension hypernym Extension.
• Abelian_extension type Software.
• Abelian_extension type Redirect.
• Abelian_extension comment "In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension.".
• Abelian_extension label "Abelian extension".
• Abelian_extension sameAs Q318705.
• Abelian_extension sameAs Abelsche_Erweiterung.
• Abelian_extension sameAs Extensión_abeliana.
• Abelian_extension sameAs Extension_abélienne.
• Abelian_extension sameAs Estensione_abeliana.
• Abelian_extension sameAs アーベル拡大.
• Abelian_extension sameAs 아벨_확대.
• Abelian_extension sameAs Abelse_uitbreiding.
• Abelian_extension sameAs Rozszerzenie_abelowe.
• Abelian_extension sameAs Extensão_abeliana.
• Abelian_extension sameAs m.017kg7.
• Abelian_extension sameAs Абелево_расширение.
• Abelian_extension sameAs Abelsk_utvidgning.
• Abelian_extension sameAs Абелеве_розширення.
• Abelian_extension sameAs Q318705.
• Abelian_extension wasDerivedFrom Abelian_extension?oldid=708273904.
• Abelian_extension isPrimaryTopicOf Abelian_extension.