About: Ramification (mathematics)

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In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping.

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rdf:type	Thing
rdfs:label	Verzweigung (Algebra) (de) Ramification (mathématiques) (fr) 분기화 (ko) 分岐 (数学) (ja) Ramification (mathematics) (en)
rdfs:comment	Verzweigung ist ein mathematischer Begriff, der die Gebiete Algebra, algebraische Geometrie, algebraische Zahlentheorie und komplexe Analysis miteinander verbindet. (de) En mathématiques, la ramification est un terme géométrique utilisé au sens de embranchement extérieur, à la façon dont la fonction racine carrée, pour les nombres complexes, peut être vue lorsqu'on considère ses deux branches opposées. Il est aussi utilisé d'une perspective opposée (branches arrivant ensemble) comme lorsqu'un revêtement dégénère en un point de la base, avec effondrement en ce point des fibres de l'application. (fr) In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. (en) ( 이 문서는 대수적 수론에서의 개념에 관한 것입니다. 복소기하학에서의 개념에 대해서는 분지점 문서를, 대수기하학에서의 개념에 대해서는 비분기 사상 문서를 참고하십시오.) 대수적 수론에서 분기화(分岐化, 영어: ramification)는 어떤 체의 확대에서, 원래 체의 대수적 정수환에서의 소 아이디얼이 확대체의 정수환에서 제곱 인자를 갖는 것을 뜻한다. 분기화는 두 가지의 방법으로 다룰 수 있다. * 분기화를 아이디얼의 소인수 분해로서 다룰 수 있다. 이는 데데킨트 정역에 대하여 적용할 수 있으나, 오직 유한 자리만을 다룰 수 있다. * 분기화를 체 위의 절댓값으로서 다룰 수 있다. 절댓값 이론은 아이디얼 이론보다 더 일반적이며, 무한 자리 또한 일관적으로 다룰 수 있다. (ko) 数学における分岐 (ramification) とは、例えば多価関数としての平方根が零点から符号の異なる二つの枝に分かれているような意味で、「枝分かれ」することをいう。またその逆に、例えばある点で退化しているような被覆写像により複数のファイバーが合流するような場合も（逆の視点から見れば枝分かれしているので）分岐という。 (ja)
rdfs:seeAlso	dbr:Galois_extensions dbr:Splitting_of_prime_ideals
foaf:depiction
dcterms:subject	Algebraic topology Complex analysis Algebraic number theory
Wikipage page ID	384644 (xsd:integer)
Wikipage revision ID	1124315153 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime ideal Euler–Poincaré characteristic Algebraic number field Relative different Richard Dedekind Riemann surface Ring of integers Valuation (algebra) Dedekind domain Degeneracy (mathematics) Complex analysis Complex number Genus (mathematics) Circle Geometry Branch point Branched covering Monodromy Nilpotent Complex manifold Newton polygon Galois extension Galois group Galois module Riemann–Hurwitz formula Algebraic geometry Algebraic number theory Algebraic topology Field (mathematics) Finite field P-adic number Knot theory Ramification group Ramification theory of valuations Complex analysis Algebraic number theory Homotopy Disk (mathematics) Manifold Square root Glossary of scheme theory Integral closure Étale morphism Unramified morphism Eisenstein polynomial Valuation theory Heinrich M. Weber Covering map Relative discriminant Puiseux expansion Extension field Extension of a valuation
Link from a Wikipage to an external page	http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf%7Cdate=18
sameAs	Ramification (mathematics) Ramification (mathematics) Ramification (mathematics) Ramification (mathematics) Ramification (mathematics) Ramification (mathematics) Ramification (mathematics) Ramification (mathematics) Ramification (mathematics)
dbp:wikiPageUsesTemplate	dbt:! dbt:Cite_book dbt:Main dbt:Other_uses dbt:See_also dbt:Short_description dbt:Visible_anchor dbt:Wiktionary dbt:Harvnb dbt:Planetmath_reference dbt:Neukirch_ANT
thumbnail	wiki-commons:Special:FilePath/Schematic_depiction_of_ramification.svg?width=300
title	Splitting and ramification in number fields and Galois extensions (en)
urlname	SplittingAndRamificationInNumberFieldsAndGaloisExtensions (en)
has abstract	Verzweigung ist ein mathematischer Begriff, der die Gebiete Algebra, algebraische Geometrie, algebraische Zahlentheorie und komplexe Analysis miteinander verbindet. (de) En mathématiques, la ramification est un terme géométrique utilisé au sens de embranchement extérieur, à la façon dont la fonction racine carrée, pour les nombres complexes, peut être vue lorsqu'on considère ses deux branches opposées. Il est aussi utilisé d'une perspective opposée (branches arrivant ensemble) comme lorsqu'un revêtement dégénère en un point de la base, avec effondrement en ce point des fibres de l'application. (fr)

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