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 noncommutative 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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© 2018 Springer-Verlag GmbH Germany, part of Springer Nature
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Aigner, M., Ziegler, G.M. (2018). Every finite division ring is a field. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57265-8_6
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DOI: https://doi.org/10.1007/978-3-662-57265-8_6
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-57264-1
Online ISBN: 978-3-662-57265-8
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