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+ | : ''For the field in relations, see [[field (relation)]].'' |
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− | A field is a [[set]] paired with two [[operation|operations]] on the set, which are designated as addition (<math>+</math>) and multiplication (<math>\cdot</math>). As a [[group]] can be conceptualized as an [[tuple|ordered pair]] of a set and an operation, <math>\left(G,\cdot\right)</math>, a field can be conceptualized as an ordered triple <math>\left(F,+,\cdot\right)</math>. |
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− | A set with addition and multiplication |
+ | A '''field''' is a [[set]] paired with two [[operation]]s on the [[set]], which are designated as addition <math>(+)</math> and multiplication <math>(\cdot)</math> . As a [[group]] can be conceptualized as an [[ordered pair]] of a set and an operation, <math>(G,\cdot)</math> , a field can be conceptualized as an ordered triple <math>(F,+,\cdot)</math> . |
+ | |||
⚫ | |||
− | + | A set with addition and multiplication, <math>(F,+,\cdot)</math> , is a field if and only if it satisfies the following properties: |
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− | # |
+ | #[[Commutative property|Commutativity]] of both addition and multiplication — [[universal quantification|For all]] <math>a,b\in F</math> , <math>a+b=b+a</math> and <math>a\cdot b=b\cdot a</math> |
− | # |
+ | #[[Associative property|Associativity]] of both addition and multiplication — For all <math>a,b,c\in F</math> , <math>(a+b)+c=a+(b+c)</math> and <math>(a\cdot b)\cdot c=a\cdot(b\cdot c)</math> |
− | # |
+ | #Additive [[Identity element|Identity]] — [[existential quantification|There exists]] a "zero" element, <math>0\in F</math> , called an additive identity, such that <math>a+0=a</math> for all <math>a\in F</math> |
− | # |
+ | #Additive [[Inverse]]s — For each <math>a\in F</math>, there exists a <math>b\in F</math> , called an additive inverse of <math>a</math> , such that <math>a+b=0</math> |
− | #[[ |
+ | #Multiplicative [[Identity element|Identity]] — There exists a "one" element, <math>1\in F</math>, different from 0, called a multiplicative identity, such that <math>a\cdot1=a</math> for all <math>a\in F</math> |
− | #[[ |
+ | #Multiplicative [[Inverse]]s — For each <math>a\in F</math> , except for 0, there exists a <math>c\in F</math> , called a multiplicative inverse of <math>a</math> , such that <math>a\cdot c=1</math> |
+ | #[[Distributive property]] — For all <math>a,b,c\in F</math> , <math>a\cdot(b+c)=a\cdot b+a\cdot c</math> |
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⚫ | |||
Alternatively, a field can be defined as a [[commutative property|commutative]] [[ring]] with unity (has a multiplicative identity) and multiplicative inverses. |
Alternatively, a field can be defined as a [[commutative property|commutative]] [[ring]] with unity (has a multiplicative identity) and multiplicative inverses. |
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− | We will often abbreviate the multiplication of two elements, <math>a |
+ | We will often abbreviate the multiplication of two elements, <math>a\cdot b</math> , by juxtaposition of the elements, <math>ab</math> . Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in [[parenthesis]]. That is, <math>a+bc+d=a+(bc)+d</math> . |
− | We can also denote <math>-a</math> and <math>a^{-1}</math> as additive and multiplicative inverses of any <math>a |
+ | We can also denote <math>-a</math> and <math>a^{-1}</math> as additive and multiplicative inverses of any <math>a\in F</math> . Furthermore, we can define two more operations, called [[subtraction]] and [[division]] by <math>a-b=a+(-b)</math> , and provided that <math>b\ne0</math> , <math>\frac{a}{b}=a\cdot b^{-1}</math> . |
==Important Results== |
==Important Results== |
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Because a field is also a ring with unity, these properties are inherited: |
Because a field is also a ring with unity, these properties are inherited: |
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− | *<math> |
+ | *<math>(F,+)</math> is an [[abelian groups]] |
− | *<math>0 |
+ | *<math>0\cdot a=0</math> , for all <math>a\in F</math> |
− | *<math>a |
+ | *<math>a(-b)=(-a)b=-(ab)</math> , for all <math>a,b\in F</math> |
− | *<math> |
+ | *<math>(-a)(-b)=ab</math> , for all <math>a,b\in F</math> |
− | *<math> |
+ | *<math>(-1)\cdot a=-a</math> , for all <math>a\in F</math> |
− | *<math> |
+ | *<math>(-1)\cdot(-1)=1</math> |
*Multiplication distributes over subtraction. |
*Multiplication distributes over subtraction. |
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Additionally: |
Additionally: |
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− | *<math> |
+ | *<math>(F^*,\cdot)</math> is also an abelian group, where <math>F^*</math> is the set of nonzero elements of <math>F</math> |
− | *Any field contains a subfield <math>K</math> that is field-isomorphic to <math>\ |
+ | *Any field contains a subfield <math>K</math> that is field-isomorphic to <math>\Q</math> or <math>\Z_p</math> for some prime <math>p</math> . |
==Optional Properties== |
==Optional Properties== |
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− | A field <math> |
+ | A field <math>(F,+,\cdot)</math> is: |
− | *A [[subfield]] of a field <math> |
+ | *A [[subfield]] of a field <math>(K,+,\cdot)</math> if <math>F\subseteq K</math> (see [[subset]]), where addition and multiplication on <math>F</math> is a [[domain restriction]] on the addition and multiplication on <math>K</math> . More commonly, we say that <math>K</math> is an [[extension field]] of <math>F</math> , and in fact, is also a [[vector space]] over <math>F</math> |
− | *An [[ordered field]] if there exists a [[total order]] <math>\le</math> on <math>F</math> such that for all <math>a, |
+ | *An [[ordered field]] if there exists a [[total order]] <math>\le</math> on <math>F</math> such that for all <math>a,b,c\in G</math> , if <math>a\le b</math>, then <math>a+c\le b+c</math> , ([[translation invariance]]), and if <math>0\le a</math> and <math>0 \le b</math> , then <math>0\le ab</math> |
==Examples== |
==Examples== |
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− | *Under the usual operations of addition and multiplication, the [[rational number]]s (<math>\ |
+ | *Under the usual operations of addition and multiplication, the [[rational number]]s (<math>\Q</math>), [[algebraic number]]s (<math>\mathbb A</math>), [[real number]]s (<math>\R</math>), and [[complex number]]s (<math>\C</math>) are fields. |
− | *An [[extension field]] of <math>\ |
+ | *An [[extension field]] of <math>\Q</math> , such as <math>\Q\left[\sqrt2\right]=\left\{a+b\sqrt2\mid a,b\in\Q\right\}</math> . |
+ | ==Related== |
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+ | Elements of a field are the quantities over the [[vectorspace]]s are constructed and there are also called the [[scalar]]s. |
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+ | In the same branch functions <math>X\to F</math> , where <math>F</math> is a field are called [[scalar field]]s. |
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[[Category:Algebra]] |
[[Category:Algebra]] |
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+ | [[Category:Linear algebra]] |
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+ | [[Category:Tensors]] |
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+ | [[Category:Differential geometry]] |
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+ | [[Category:Abstract algebra]] |
Latest revision as of 23:34, 21 March 2017
- For the field in relations, see field (relation).
A field is a set paired with two operations on the set, which are designated as addition and multiplication . As a group can be conceptualized as an ordered pair of a set and an operation, , a field can be conceptualized as an ordered triple .
A set with addition and multiplication, , is a field if and only if it satisfies the following properties:
- Commutativity of both addition and multiplication — For all , and
- Associativity of both addition and multiplication — For all , and
- Additive Identity — There exists a "zero" element, , called an additive identity, such that for all
- Additive Inverses — For each , there exists a , called an additive inverse of , such that
- Multiplicative Identity — There exists a "one" element, , different from 0, called a multiplicative identity, such that for all
- Multiplicative Inverses — For each , except for 0, there exists a , called a multiplicative inverse of , such that
- Distributive property — For all ,
- Closure of addition and multiplication — For all , and
Alternatively, a field can be defined as a commutative ring with unity (has a multiplicative identity) and multiplicative inverses.
We will often abbreviate the multiplication of two elements, , by juxtaposition of the elements, . Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, .
We can also denote and as additive and multiplicative inverses of any . Furthermore, we can define two more operations, called subtraction and division by , and provided that , .
Important Results
Because a field is also a ring with unity, these properties are inherited:
- is an abelian groups
- , for all
- , for all
- , for all
- , for all
- Multiplication distributes over subtraction.
Additionally:
- is also an abelian group, where is the set of nonzero elements of
- Any field contains a subfield that is field-isomorphic to or for some prime .
Optional Properties
A field is:
- A subfield of a field if (see subset), where addition and multiplication on is a domain restriction on the addition and multiplication on . More commonly, we say that is an extension field of , and in fact, is also a vector space over
- An ordered field if there exists a total order on such that for all , if , then , (translation invariance), and if and , then
Examples
- Under the usual operations of addition and multiplication, the rational numbers (), algebraic numbers (), real numbers (), and complex numbers () are fields.
- An extension field of , such as .
Related
Elements of a field are the quantities over the vectorspaces are constructed and there are also called the scalars.
In the same branch functions , where is a field are called scalar fields.