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- Separable_extension abstract "In the subfield of algebra named field theory, a separable extension is an algebraic field extension such that for every , the minimal polynomial of over F is a separable polynomial (i.e., has distinct roots; see below for the definition in this context). Otherwise, the extension is called inseparable. There are other equivalent definitions of the notion of a separable algebraic extension, and these are outlined later in the article. The importance of separable extensions lies in the fundamental role they play in Galois theory in finite characteristic. More specifically, a finite degree field extension is Galois if and only if it is both normal and separable. Since algebraic extensions of fields of characteristic zero, and of finite fields, are separable, separability is not an obstacle in most applications of Galois theory. For instance, every algebraic (in particular, finite degree) extension of the field of rational numbers is necessarily separable.Despite the ubiquity of the class of separable extensions in mathematics, its extreme opposite, namely the class of purely inseparable extensions, also occurs quite naturally. An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial (i.e., does not have distinct roots). For a field F to possess a non-trivial purely inseparable extension, it must necessarily be an infinite field of prime characteristic (i.e. specifically, imperfect), since any algebraic extension of a perfect field is necessarily separable.".
- Separable_extension wikiPageExternalLink ISBN978-4-7853-1309-8.htm.
- Separable_extension wikiPageID "363340".
- Separable_extension wikiPageRevisionID "598007793".
- Separable_extension hasPhotoCollection Separable_extension.
- Separable_extension id "s/s084470".
- Separable_extension title "separable extension of a field k".
- Separable_extension subject Category:Field_extensions.
- Separable_extension type Abstraction100002137.
- Separable_extension type Delay115272029.
- Separable_extension type Extension115272382.
- Separable_extension type FieldExtensions.
- Separable_extension type Measure100033615.
- Separable_extension type Pause115271008.
- Separable_extension type TimeInterval115269513.
- Separable_extension comment "In the subfield of algebra named field theory, a separable extension is an algebraic field extension such that for every , the minimal polynomial of over F is a separable polynomial (i.e., has distinct roots; see below for the definition in this context). Otherwise, the extension is called inseparable. There are other equivalent definitions of the notion of a separable algebraic extension, and these are outlined later in the article.".
- Separable_extension label "Estensione separabile".
- Separable_extension label "Extension séparable".
- Separable_extension label "Extensión separable".
- Separable_extension label "Extensão separável".
- Separable_extension label "Separable extension".
- Separable_extension label "Сепарабельное расширение".
- Separable_extension label "可分扩张".
- Separable_extension sameAs Extensión_separable.
- Separable_extension sameAs Extension_séparable.
- Separable_extension sameAs Estensione_separabile.
- Separable_extension sameAs Extensão_separável.
- Separable_extension sameAs m.01_pmv.
- Separable_extension sameAs Q2264756.
- Separable_extension sameAs Q2264756.
- Separable_extension sameAs Separable_extension.
- Separable_extension wasDerivedFrom Separable_extension?oldid=598007793.
- Separable_extension isPrimaryTopicOf Separable_extension.