Difference between revisions of "Field"
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A '''field''' is a structure of [[abstract algebra]], similar to a [[group]] or a [[ring]]. A field <math>F</math> is a set of elements with two operations, usually called multiplication and addition and denoted <math>\cdot</math> and <math>+</math>, which have the following properties: | A '''field''' is a structure of [[abstract algebra]], similar to a [[group]] or a [[ring]]. A field <math>F</math> is a set of elements with two operations, usually called multiplication and addition and denoted <math>\cdot</math> and <math>+</math>, which have the following properties: | ||
| − | + | A field is a ring. Thus, a field obeys all of the ring axioms. | |
| + | |||
| + | <math>1 \neq 0</math>. | ||
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| + | If we exclude 0, the remaining elements form an [[abelian group]] under the operation <math>\cdot</math>. In particular, multiplicitive inverses exist for every element other than 0. | ||
Common examples of fields are the [[rational number]]s, the [[real number]]s or the [[integer]]s taken [[modulo]] some [[prime]]. In each case, addition and multiplication are "as usual." | Common examples of fields are the [[rational number]]s, the [[real number]]s or the [[integer]]s taken [[modulo]] some [[prime]]. In each case, addition and multiplication are "as usual." | ||
Revision as of 10:43, 16 July 2006
A field is a structure of abstract algebra, similar to a group or a ring. A field 
is a set of elements with two operations, usually called multiplication and addition and denoted 
and 
, which have the following properties:
A field is a ring. Thus, a field obeys all of the ring axioms.
.
If we exclude 0, the remaining elements form an abelian group under the operation . In particular, multiplicitive inverses exist for every element other than 0.
Common examples of fields are the rational numbers, the real numbers or the integers taken modulo some prime. In each case, addition and multiplication are "as usual."
