Changes

== Axioms ==

== Axioms ==

Technically, a field F consists of a set of elements and two [[Binary operation|binary operations]], addition + and multiplication *, which obey the following axioms:

Technically, a field <math>F</math> is a set of elements endowed with two [[Binary operation|binary operations]], <math>+</math> and <math>\cdot</math> (with properties analogous to addition and multiplication, respectively), which obey the following axioms:

'''Addition'''

*Addition axioms

*#Closure: If <math>a, b \in F</math> then <math>a + b \in F</math>.

1. Closure: If a,b ∈ F then a + b ∈ F.

*#[[Associative property|Associativity]]: For <math>a, b, c \in F<math>, <math>(a + b) + c = a + (b + c)</math>.

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*#[[Commutative property|Commutativity]]: For <math>a, b \in F</math> <math>a + b = b + a</math>.

2. [[Associative property|Associativity]]: For a,b,c ∈ F, (a + b) + c = a + (b + c).

*#[[Identity Element|Identity]]: There exists an element <math>0 \in F</math> such that <math>a + 0 = 0 + a = a</math> for all <math>a \in F</math>.

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*#[[Inverse]]: For all <math>a \in F</math> there exists an element <math>-a \in F</math> such that <math>a + (-a) = (-a) + a = 0</math>.

3. [[Commutative property|Commutativity]]: For a,b ∈ F,  a + b = b + a

*Multiplication axioms

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*#Closure: If <math>a, b \in F</math> then <math>a \cdot b \in F</math>.

4.  [[Identity_Element|Identity]]: There exists an element 0 ∈ F such that a + 0 = 0 + a = a for all a ∈ F.

*#[[Associative property|Associativity]]: For <math>a, b, c \in F</math>, <math>(a \cdot b) \cdot c = a \cdot (b \cdot c)</math>.

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*#[[Commutative property|Commutativity]]: For <math>a, b \in F</math>, <math>a \cdot b = b \cdot a</math>.

5. [[Inverse]]: If a ∈ F then there exists an (-a) ∈ F such that a + (-a) = (-a) + a = 0.

*#[[Identity Element|Identity]]: There exists an element <math>1 \in F</math> such that <math>a \cdot 1 = 1 \cdot a = a</math> for all <math>a \in F</math>.

 

*#[[Inverse]]: If <math>a \in F</math> and <math>a \neq 0</math> then there exists an element <math>a^{-1} \in F</math> such that <math>a \cdot a^{-1} = a^{-1} \cdot a = 1</math>.

'''Multiplication'''

*Distributivity: For <math>a, b, c \in F</math>, <math>a \cdot (b + c) = (a \cdot b) + (a \cdot c)</math>.

1. Closure: If a,b ∈ F then a*b ∈ F.

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2. [[Associative property|Associativity]]: For a,b,c ∈ F. (a*b)*c = a*(b*c).

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3. [[Commutative property|Commutativity]]: For a,b ∈ F, a*b = b*a

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4.  [[Identity_Element|Identity]]: There exists an element 1 ∈ F such that a*1 = 1*a = a for all a ∈ F.

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5. [[Inverse]]: If a ∈ F and a ≠ 0 then there exists an a^-1 ∈ F such that a*a^-1 = a^-1*a = 1.

 

'''Distributivity'''

 

For a,b,c ∈ F, a*(b + c) = a*b + a*c.  

Sometimes the condition 0 ≠ 1 is also included.

Sometimes the condition 0 ≠ 1 is also included.

[[Category:Algebra]]

[[Category:Algebra]]