Every finite division ring is a field
Identifieur interne : 000430 ( Main/Exploration ); précédent : 000429; suivant : 000431Every 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. This result which is now a classic has caught the imagination of many mathematicians, because, as Herstein writes: “It is so unexpectedly interrelating two seemingly unrelated things, the number of elements in a certain algebraic system and the multiplication of that system.”
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DOI: 10.1007/978-3-642-00856-6_6
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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></author><author><name sortKey="Ziegler, Gunter M" sort="Ziegler, Gunter M" uniqKey="Ziegler G" first="Günter M." last="Ziegler">Günter M. Ziegler</name></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:30507D3439332AE71ED7C9AAFA472A0B7F4992EF</idno><date when="2010" year="2010">2010</date><idno type="doi">10.1007/978-3-642-00856-6_6</idno><idno type="url">https://api.istex.fr/document/30507D3439332AE71ED7C9AAFA472A0B7F4992EF/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">000A17</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Corpus" wicri:corpus="ISTEX">000A17</idno><idno type="wicri:Area/Istex/Curation">000A17</idno><idno type="wicri:Area/Istex/Checkpoint">000386</idno><idno type="wicri:explorRef" wicri:stream="Istex" wicri:step="Checkpoint">000386</idno><idno type="wicri:Area/Main/Merge">000430</idno><idno type="wicri:Area/Main/Curation">000430</idno><idno type="wicri:Area/Main/Exploration">000430</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="3"><country xml:lang="fr">Allemagne</country><wicri:regionArea>FU Berlin, Berlin</wicri:regionArea><placeName><region type="land" nuts="3">Berlin</region><settlement type="city">Berlin</settlement></placeName></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="3"><country xml:lang="fr">Allemagne</country><wicri:regionArea>FU Berlin, Berlin</wicri:regionArea><placeName><region type="land" nuts="3">Berlin</region><settlement type="city">Berlin</settlement></placeName></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. This result which is now a classic has caught the imagination of many mathematicians, because, as Herstein writes: “It is so unexpectedly interrelating two seemingly unrelated things, the number of elements in a certain algebraic system and the multiplication of that system.”</div></front></TEI><affiliations><list><country><li>Allemagne</li></country><region><li>Berlin</li></region><settlement><li>Berlin</li></settlement></list><tree><country name="Allemagne"><region name="Berlin"><name sortKey="Aigner, Martin" sort="Aigner, Martin" uniqKey="Aigner M" first="Martin" last="Aigner">Martin Aigner</name></region><name sortKey="Ziegler, Gunter M" sort="Ziegler, Gunter M" uniqKey="Ziegler G" first="Günter M." last="Ziegler">Günter M. Ziegler</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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