Every finite division ring is a field
Identifieur interne : 001F12 ( Istex/Curation ); précédent : 001F11; suivant : 001F13Every finite division ring is a field
Auteurs : Martin Aigner [Allemagne] ; Günter M. Ziegler [Allemagne]Source :
Abstract
Abstract: Rings are important structures in modern algebra. If a ring R has a multiplicative unit element 1 and every nonzero element has a multiplicative inverse, then R is called a division ring. So, all that is missing in R from being a field is the commutativity of multiplication. The best-known example of a non-commutative division ring is the ring of quaternions discovered by Hamilton. But, as the chapter title says, every such division ring must of necessity be infinite. If R is finite, then the axioms force the multiplication to be commutative.
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DOI: 10.1007/978-3-662-22343-7_5
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Every finite division ring is a field</title><author><name sortKey="Aigner, Martin" sort="Aigner, Martin" uniqKey="Aigner M" first="Martin" last="Aigner">Martin Aigner</name><affiliation wicri:level="1"><mods:affiliation>Institut für Mathematik II (WE2), Freie Universität Berlin, Arnimallee 3, D-14195, Berlin, Germany</mods:affiliation><country xml:lang="fr">Allemagne</country><wicri:regionArea>Institut für Mathematik II (WE2), Freie Universität Berlin, Arnimallee 3, D-14195, Berlin</wicri:regionArea></affiliation><affiliation wicri:level="1"><mods:affiliation>E-mail: aigner@math.fu-berlin.de</mods:affiliation><country wicri:rule="url">Allemagne</country></affiliation></author><author><name sortKey="Ziegler, Gunter M" sort="Ziegler, Gunter M" uniqKey="Ziegler G" first="Günter M." last="Ziegler">Günter M. Ziegler</name><affiliation wicri:level="1"><mods:affiliation>Fachbereich Mathematik, MA 7-1, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin, Germany</mods:affiliation><country xml:lang="fr">Allemagne</country><wicri:regionArea>Fachbereich Mathematik, MA 7-1, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin</wicri:regionArea></affiliation><affiliation wicri:level="1"><mods:affiliation>E-mail: ziegler@math.tu-berlin.de</mods:affiliation><country wicri:rule="url">Allemagne</country></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:98555404F425F6DEB683F308B007D0B929119F76</idno><date when="1998" year="1998">1998</date><idno type="doi">10.1007/978-3-662-22343-7_5</idno><idno type="url">https://api.istex.fr/document/98555404F425F6DEB683F308B007D0B929119F76/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001F12</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">001F12</idno><idno type="wicri:Area/Istex/Curation">001F12</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Every finite division ring is a field</title><author><name sortKey="Aigner, Martin" sort="Aigner, Martin" uniqKey="Aigner M" first="Martin" last="Aigner">Martin Aigner</name><affiliation wicri:level="1"><mods:affiliation>Institut für Mathematik II (WE2), Freie Universität Berlin, Arnimallee 3, D-14195, Berlin, Germany</mods:affiliation><country xml:lang="fr">Allemagne</country><wicri:regionArea>Institut für Mathematik II (WE2), Freie Universität Berlin, Arnimallee 3, D-14195, Berlin</wicri:regionArea></affiliation><affiliation wicri:level="1"><mods:affiliation>E-mail: aigner@math.fu-berlin.de</mods:affiliation><country wicri:rule="url">Allemagne</country></affiliation></author><author><name sortKey="Ziegler, Gunter M" sort="Ziegler, Gunter M" uniqKey="Ziegler G" first="Günter M." last="Ziegler">Günter M. Ziegler</name><affiliation wicri:level="1"><mods:affiliation>Fachbereich Mathematik, MA 7-1, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin, Germany</mods:affiliation><country xml:lang="fr">Allemagne</country><wicri:regionArea>Fachbereich Mathematik, MA 7-1, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin</wicri:regionArea></affiliation><affiliation wicri:level="1"><mods:affiliation>E-mail: ziegler@math.tu-berlin.de</mods:affiliation><country wicri:rule="url">Allemagne</country></affiliation></author></analytic><monogr></monogr></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: Rings are important structures in modern algebra. If a ring R has a multiplicative unit element 1 and every nonzero element has a multiplicative inverse, then R is called a division ring. So, all that is missing in R from being a field is the commutativity of multiplication. The best-known example of a non-commutative division ring is the ring of quaternions discovered by Hamilton. But, as the chapter title says, every such division ring must of necessity be infinite. If R is finite, then the axioms force the multiplication to be commutative.</div></front></TEI></record>Pour manipuler ce document sous Unix (Dilib)
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