23/03/19 10:36:40.33 7NhejE26.net
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URLﾘﾝｸ(en.wikipedia.org)(mathematics)
Field (mathematics)
Related notions
Division rings
Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all field axioms except for the existence of multiplicative inverses. Dropping instead commutativity of multiplication leads to the concept of a division ring or skew field;[nb 7] sometimes associativity is weakened as well. The only division rings that are finite-dimensional R-vector spaces are R itself, C (which is a field), and the quaternions H (in which multiplication is non-commutative). This result is known as the Frobenius theorem. The octonions O, for which multiplication is neither commutative nor associative, is a normed alternative division algebra, but is not a division ring. This fact was proved using methods of algebraic topology in 1958 by Michel Kervaire, Raoul Bott, and John Milnor.[62] The non-existence of an odd-dimensional division algebra is more classical. It can be deduced from the hairy ball theorem illustrated at the right.[citation needed]
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