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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References
L. E. Dickson: On finite algebras, Nachrichten der Akad. Wissenschaften Göttingen Math.-Phys. Klasse (1905), 1–36; Collected Mathematical Papers Vol. III, Chelsea Publ. Comp, The Bronx, NY 1975, 539–574.
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E. Witt: Über die Kommutativität endlicher Schiefkörper, Abh. Math. Sem. Univ. Hamburg 8 (1931), 413.
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© 2004 Springer-Verlag Berlin Heidelberg
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Aigner, M., Ziegler, G.M. (2004). Every finite division ring is a field. In: Proofs from THE BOOK. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05412-3_5
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DOI: https://doi.org/10.1007/978-3-662-05412-3_5
Publisher Name: Springer, Berlin, Heidelberg
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