Every finite division ring is a field
Identifieur interne : 001787 ( Main/Merge ); précédent : 001786; suivant : 001788Every 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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<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>
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