Difference between revisions of "Field (mathematics)"
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| − | {{stub}}A '''field''' is a commutative [[Ring (mathematics)|ring]] which contains a non-zero multiplicative identity and all non-zero elements have multiplicative inverses. Everyday examples of fields include the [[real numbers]], [[complex numbers]] and the [[rationals]]. The [[characteristic]] of a field must be either 0 or a [[prime number]] ''p''. For each prime number ''p'' and positive integer ''n'', there is a unique (up to [[isomorphism]]) finite field of [[characteristic]] ''p'' whose [[cardinality]] is ''p<sup>n</sup>''. A field of characteristic 0 is necessarily infinite. | + | {{stub}} |
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| + | A '''field''' is a commutative [[Ring (mathematics)|ring]] which contains a non-zero multiplicative identity and all non-zero elements have multiplicative inverses. Everyday examples of fields include the [[real numbers]], [[complex numbers]] and the [[rationals]]. The [[characteristic]] of a field must be either 0 or a [[prime number]] ''p''. For each prime number ''p'' and positive integer ''n'', there is a unique (up to [[isomorphism]]) finite field of [[characteristic]] ''p'' whose [[cardinality]] is ''p<sup>n</sup>''. A field of characteristic 0 is necessarily infinite. | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
Revision as of 02:28, February 9, 2009
A field is a commutative ring which contains a non-zero multiplicative identity and all non-zero elements have multiplicative inverses. Everyday examples of fields include the real numbers, complex numbers and the rationals. The characteristic of a field must be either 0 or a prime number p. For each prime number p and positive integer n, there is a unique (up to isomorphism) finite field of characteristic p whose cardinality is pn. A field of characteristic 0 is necessarily infinite.
