Difference between revisions of "Field"
ComplexZeta (talk | contribs) m |
m |
||
| Line 1: | Line 1: | ||
| − | 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 | + | 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 [[operation]]s, 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. | * A field is a ring. Thus, a field obeys all of the ring axioms. | ||
| − | * <math>1 \neq 0</math>. | + | * <math>1 \neq 0</math>, where 1 is the multiplicative [[identity]] and 0 is the additive indentity. Thus fields have at least 2 elements. |
| − | * 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. | + | * If we exclude 0, the remaining elements form an [[abelian group]] under the operation <math>\cdot</math>. In particular, multiplicitive [[inverse with respect to an operation | 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 08:30, 18 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.
, where 1 is the multiplicative identity and 0 is the additive indentity. Thus fields have at least 2 elements.- 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.
, where 1 is the multiplicative
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."
