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''' is a structure of [[abstract algebra]], similar to a [[group]] or a [[ring]]. Informally, fields are the general structure in which the usual laws of [[arithmetic]] governing the operations <math>+, -, \times</math> and <math>\div</math> hold. In particular, the [[rational number]]s, the [[real number]]s and the [[complex number]]s are all fields. |
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| + | Formally, a field <math>F</math> is a [[set]] of elements with two [[operation]]s, usually called multiplication and addition (denoted <math>\cdot</math> and <math>+</math>, respectively) 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>, where 1 is the multiplicative [[identity]] and 0 is the additive indentity. Thus fields have at least 2 elements. | * <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 | + | * If we exclude 0, the remaining elements form an [[abelian group]] under multiplication. In particular, multiplicative [[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." Other examples include the set of [[algebraic number]]s and [[finite field]]s other than the integers modulo a prime. |
[[Category:Field theory]] | [[Category:Field theory]] | ||
Revision as of 23:11, 10 February 2011
A field is a structure of abstract algebra, similar to a group or a ring. Informally, fields are the general structure in which the usual laws of arithmetic governing the operations 
and 
hold. In particular, the rational numbers, the real numbers and the complex numbers are all fields.
Formally, a field 
is a set of elements with two operations, usually called multiplication and addition (denoted 
and 
, respectively) 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 multiplication. In particular, multiplicative 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." Other examples include the set of algebraic numbers and finite fields other than the integers modulo a prime.
