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- Algebraic_extension abstract "In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental.For example, the field extension R/Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C/R and Q(√2)/Q are algebraic, where C is the field of complex numbers.All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic. The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q[a] is an example of an algebraic number field.A field with no nontrivial algebraic extensions is called algebraically closed. An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice.An extension L/K is algebraic if and only if every sub K-algebra of L is a field.".
- Algebraic_extension wikiPageID "2125".
- Algebraic_extension wikiPageRevisionID "600609240".
- Algebraic_extension hasPhotoCollection Algebraic_extension.
- Algebraic_extension subject Category:Abstract_algebra.
- Algebraic_extension subject Category:Field_extensions.
- Algebraic_extension type Abstraction100002137.
- Algebraic_extension type Delay115272029.
- Algebraic_extension type Extension115272382.
- Algebraic_extension type FieldExtensions.
- Algebraic_extension type Measure100033615.
- Algebraic_extension type Pause115271008.
- Algebraic_extension type TimeInterval115269513.
- Algebraic_extension comment "In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e.".
- Algebraic_extension label "Algebraic extension".
- Algebraic_extension label "Algebraische Erweiterung".
- Algebraic_extension label "Algebraïsche uitbreiding".
- Algebraic_extension label "Estensione algebrica".
- Algebraic_extension label "Extension algébrique".
- Algebraic_extension label "Extensión algebraica".
- Algebraic_extension label "Extensão algébrica".
- Algebraic_extension label "Алгебраическое расширение".
- Algebraic_extension label "代數擴張".
- Algebraic_extension sameAs Algebraické_nadtěleso.
- Algebraic_extension sameAs Algebraische_Erweiterung.
- Algebraic_extension sameAs Extensión_algebraica.
- Algebraic_extension sameAs Extension_algébrique.
- Algebraic_extension sameAs Estensione_algebrica.
- Algebraic_extension sameAs 대수적_확대.
- Algebraic_extension sameAs Algebraïsche_uitbreiding.
- Algebraic_extension sameAs Extensão_algébrica.
- Algebraic_extension sameAs m.0w_3.
- Algebraic_extension sameAs Q550791.
- Algebraic_extension sameAs Q550791.
- Algebraic_extension sameAs Algebraic_extension.
- Algebraic_extension wasDerivedFrom Algebraic_extension?oldid=600609240.
- Algebraic_extension isPrimaryTopicOf Algebraic_extension.