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endpoint"> Sparql Endpoint </a> </li> </ul> </div> </div> </nav> <div style="margin-bottom: 60px"></div>   <section> <div class="container-xl"> <div class="row"> <div class="col"> <h1 id="title" class="display-6">About: <a href="http://dbpedia.org/resource/Tensor_product_of_fields">Tensor product of fields</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> An Entity of Type: <a href="http://www.w3.org/2002/07/owl#Thing">Thing</a>, from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, within Data Space: <a href="http://dbpedia.org">dbpedia.org</a> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield. The tensor product of two fields is sometimes a field, and often a direct product of fields; In some cases, it can contain non-zero nilpotent elements. The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field. </div> </div> </div> </section>   <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract">dbo:abstract</a> </td><td class="col-10 text-break"><ul> <li>In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield. The tensor product of two fields is sometimes a field, and often a direct product of fields; In some cases, it can contain non-zero nilpotent elements. The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field. (en)</li> <li style="display:none;">抽象代数学において体論には直積（いうなれば「直積体」）が存在しない（二つの体の（それらを環と見做してとった）直積（直積環）が、それ自身体になることは無いから）。その一方で、たとえば体 K と L がより大きい体 M の部分体として与えられているときや体 K と L が両方より小さい体 N（例えば素体）の拡大体のときには、その二つの体 K と L を「併せる」ことがしばしば要求される。そういった体の間で生じるすべての現象を議論するために利用できる、それら体上の構成として体のテンソル積 (tensor product of fields) は最善である。これは環としてのテンソル積（テンソル積環）であり（それ自体、環にはなるが）、体になることもあれば体の直積環となることも多い。その一方で、0 でない冪零元を含みうる（環の根基参照）。体 K と L が同型な素体を持たなければ ―つまり標数が異なれば― ある体 M の共通の部分体では決してない。このことに対応するのは「体 K と L のテンソル積が自明環になる」ことである。（このようにテンソル積構成が潰れてしまうのは理論としてはつまらない内容しか含まないので、ここでは特に扱わない） (ja)</li> <li style="display:none;">Em álgebra, o produto tensorial de corpos, de dois corpos K e L incluídos em um terceiro corpo M é o menor sub-corpo de M contendo tanto K e L. (pt)</li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageExternalLink">dbo:wikiPageExternalLink</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="http://www.jmilne.org/math/CourseNotes/ANT.pdf" href="http://www.jmilne.org/math/CourseNotes/ANT.pdf">http://www.jmilne.org/math/CourseNotes/ANT.pdf</a></li> <li><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="http://abel.math.harvard.edu/archive/129_spring_04/ant/ant.pdf" href="http://abel.math.harvard.edu/archive/129_spring_04/ant/ant.pdf">http://abel.math.harvard.edu/archive/129_spring_04/ant/ant.pdf</a></li> <li><a class="uri" rel="dbo:wikiPageExternalLink nofollow" resource="http://mathoverflow.net/questions/8324/what-does-linearly-disjoint-mean-for-abstract-field-extensions" 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href="http://dbpedia.org/property/title">dbp:title</a> </td><td class="col-10 text-break"><ul> <li>Compositum of field extensions (en)</li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/property/wikiPageUsesTemplate">dbp:wikiPageUsesTemplate</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Springer" href="http://dbpedia.org/resource/Template:Springer">dbt:Springer</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Cite_book" href="http://dbpedia.org/resource/Template:Cite_book">dbt:Cite_book</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Cite_web" href="http://dbpedia.org/resource/Template:Cite_web">dbt:Cite_web</a></li> <li><a class="uri" rel="dbp:wikiPageUsesTemplate" resource="http://dbpedia.org/resource/Template:Distinguish" 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text-break"><ul> <li><a class="uri" rel="dcterms:subject" resource="http://dbpedia.org/resource/Category:Field_(mathematics)" prefix="dcterms: http://purl.org/dc/terms/" href="http://dbpedia.org/resource/Category:Field_(mathematics)">dbc:Field_(mathematics)</a></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/1999/02/22-rdf-syntax-ns#type">rdf:type</a> </td><td class="col-10 text-break"><ul> <li><a class="uri" rel="rdf:type" resource="http://www.w3.org/2002/07/owl#Thing" href="http://www.w3.org/2002/07/owl#Thing">owl:Thing</a></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment">rdfs:comment</a> </td><td class="col-10 text-break"><ul> <li>In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield. The tensor product of two fields is sometimes a field, and often a direct product of fields; In some cases, it can contain non-zero nilpotent elements. The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field. (en)</li> <li style="display:none;">抽象代数学において体論には直積（いうなれば「直積体」）が存在しない（二つの体の（それらを環と見做してとった）直積（直積環）が、それ自身体になることは無いから）。その一方で、たとえば体 K と L がより大きい体 M の部分体として与えられているときや体 K と L が両方より小さい体 N（例えば素体）の拡大体のときには、その二つの体 K と L を「併せる」ことがしばしば要求される。そういった体の間で生じるすべての現象を議論するために利用できる、それら体上の構成として体のテンソル積 (tensor product of fields) は最善である。これは環としてのテンソル積（テンソル積環）であり（それ自体、環にはなるが）、体になることもあれば体の直積環となることも多い。その一方で、0 でない冪零元を含みうる（環の根基参照）。体 K と L が同型な素体を持たなければ ―つまり標数が異なれば― ある体 M の共通の部分体では決してない。このことに対応するのは「体 K と L のテンソル積が自明環になる」ことである。（このようにテンソル積構成が潰れてしまうのは理論としてはつまらない内容しか含まないので、ここでは特に扱わない） (ja)</li> <li style="display:none;">Em álgebra, o produto tensorial de corpos, de dois corpos K e L incluídos em um terceiro corpo M é o menor sub-corpo de M contendo tanto K e L. 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