Difference between revisions of "Field"
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| − | 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. | + | 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 <math>\mathbb{Q}</math>, the [[real number]]s <math>\mathbb{R}</math>, and the [[complex number]]s <math>\mathbb{C}</math> are all fields. |
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: | 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: | ||
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* 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. | * 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 <math>\mathbb{Q}</math>, the [[real number]]s <math>\mathbb{R}</math>, or the [[integer]]s <math>\mathbb{Z}</math> taken [[modulo]] some [[prime]] <math>p</math>, denoted <math>\mathbb{F}_{p}</math> or <math>\mathbb{Z}/p\mathbb{Z}</math>. In each case, addition and multiplication are defined "as usual." Other examples include the set of [[algebraic number]]s and [[finite field]]s of order <math>p^{k}</math> for <math>k</math> an arbitrary positive integer. | |
| − | 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 | ||
[[Category:Field theory]] | [[Category:Field theory]] | ||
Revision as of 16:49, 16 March 2012
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 
, denoted 
or 
. In each case, addition and multiplication are defined "as usual." Other examples include the set of algebraic numbers and finite fields of order 
for 
an arbitrary positive integer.
