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Matches in DBpedia 2016-04 for { <http://dbpedia.org/resource/Algebraic_extension> ?p ?o }

• 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. The converse is true as well, if K[a] is a field, then a is algebraic 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".
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• Algebraic_extension wikiPageOutDegree "31".
• Algebraic_extension wikiPageRevisionID "653316776".
• Algebraic_extension wikiPageWikiLink Abstract_algebra.
• Algebraic_extension wikiPageWikiLink Algebraic_closure.
• Algebraic_extension wikiPageWikiLink Algebraic_element.
• Algebraic_extension wikiPageWikiLink Algebraic_number.
• Algebraic_extension wikiPageWikiLink Algebraic_number_field.
• Algebraic_extension wikiPageWikiLink Algebraically_closed_field.
• Algebraic_extension wikiPageWikiLink Axiom_of_choice.
• Algebraic_extension wikiPageWikiLink Category:Field_extensions.
• Algebraic_extension wikiPageWikiLink Complex_number.
• Algebraic_extension wikiPageWikiLink Degree_of_a_field_extension.
• Algebraic_extension wikiPageWikiLink Field_(mathematics).
• Algebraic_extension wikiPageWikiLink Field_extension.
• Algebraic_extension wikiPageWikiLink Galois_extension.
• Algebraic_extension wikiPageWikiLink Galois_group.
• Algebraic_extension wikiPageWikiLink Glossary_of_field_theory.
• Algebraic_extension wikiPageWikiLink If_and_only_if.
• Algebraic_extension wikiPageWikiLink Integral_element.
• Algebraic_extension wikiPageWikiLink Lxc3xbcroths_theorem.
• Algebraic_extension wikiPageWikiLink Model_theory.
• Algebraic_extension wikiPageWikiLink Normal_extension.
• Algebraic_extension wikiPageWikiLink Polynomial.
• Algebraic_extension wikiPageWikiLink Rational_number.
• Algebraic_extension wikiPageWikiLink Real_number.
• Algebraic_extension wikiPageWikiLink Separable_extension.
• Algebraic_extension wikiPageWikiLink Tensor_product_of_fields.
• Algebraic_extension wikiPageWikiLink Zero_of_a_function.
• Algebraic_extension wikiPageWikiLink Zorns_lemma.
• Algebraic_extension wikiPageWikiLinkText "Algebraic extension".
• Algebraic_extension wikiPageWikiLinkText "Algebraic_extension".
• Algebraic_extension wikiPageWikiLinkText "algebraic extension".
• Algebraic_extension wikiPageWikiLinkText "algebraic field extension".
• Algebraic_extension wikiPageWikiLinkText "algebraic over".
• Algebraic_extension wikiPageWikiLinkText "algebraic".
• Algebraic_extension wikiPageWikiLinkText "extends".
• Algebraic_extension wikiPageWikiLinkText "extension".
• Algebraic_extension wikiPageWikiLinkText "finite extension".
• Algebraic_extension chapter "V.1:Algebraic Extensions".
• Algebraic_extension edition "3".
• Algebraic_extension pages "223".
• Algebraic_extension wikiPageUsesTemplate Template:Citation.
• Algebraic_extension wikiPageUsesTemplate Template:Lang_Algebra.
• Algebraic_extension wikiPageUsesTemplate Template:Main.
• Algebraic_extension wikiPageUsesTemplate Template:Ordered_list.
• Algebraic_extension wikiPageUsesTemplate Template:Refimprove.
• Algebraic_extension subject Category:Field_extensions.
• Algebraic_extension hypernym Root.
• Algebraic_extension type Eukaryote.
• 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 sameAs Q550791.
• Algebraic_extension sameAs Extensió_algebraica.
• Algebraic_extension sameAs Algebraické_nadtěleso.
• Algebraic_extension sameAs Algebraische_Erweiterung.
• Algebraic_extension sameAs Extensión_algebraica.
• Algebraic_extension sameAs Algebrallinen_laajennus.
• Algebraic_extension sameAs Extension_algébrique.
• Algebraic_extension sameAs Estensione_algebrica.
• Algebraic_extension sameAs 代数拡大.
• Algebraic_extension sameAs 대수적_확대.
• Algebraic_extension sameAs Algebraïsche_uitbreiding.
• Algebraic_extension sameAs Extensão_algébrica.
• Algebraic_extension sameAs m.0w_3.
• Algebraic_extension sameAs Алгебраическое_расширение.
• Algebraic_extension sameAs Алгебричне_розширення.
• Algebraic_extension sameAs Q550791.
• Algebraic_extension sameAs 代數擴張.
• Algebraic_extension wasDerivedFrom Algebraic_extension?oldid=653316776.
• Algebraic_extension isPrimaryTopicOf Algebraic_extension.