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A '''field''' is a commutative [[Ring (mathematics)|ring]] which contains a non-zero multiplicative identity and all non-zero elements have multiplicative inverses. Loosely, a field is a collection of entities with well-behaved and compatible addition and multiplication operations. A few examples serve to illustrate this point.
A '''field''' is a commutative [[Ring (mathematics)|ring]] which contains a non-zero multiplicative identity and all non-zero elements have multiplicative inverses. Loosely, a field is a collection of entities with well-behaved and compatible addition and multiplication operations. A few examples serve to illustrate this point.
== Examples ==
== Examples ==
Fields play an important role in nearly every area of [[mathematics]], and are one of the most basic objects studied by [[algebra]]. The study of the relationships between different fields, and in particular subfields of a given field, leads to the study of [[Galois]] theory, and makes possible the proof of [[Abel-Ruffini theorem|Abel's theorem]] and was one of the motivations for the early study of fields and [[abstract algebra]] more generally.
Fields play an important role in nearly every area of [[mathematics]], and are one of the most basic objects studied by [[algebra]]. The study of the relationships between different fields, and in particular subfields of a given field, leads to the study of [[Galois]] theory, and makes possible the proof of [[Abel-Ruffini theorem|Abel's theorem]] and was one of the motivations for the early study of fields and [[abstract algebra]] more generally.
== 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:
'''Addition'''
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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 0 ∈ F such that a + 0 = 0 + a = a for all a ∈ F.
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5. [[Inverse]]: If a ∈ F then there exists an (-a) ∈ F such that a + (-a) = (-a) + a = 0.
'''Multiplication'''
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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<sup>-1</sup> ∈ F such that a*a<sup>-1</sup> = a<sup>-1</sup>*a = 1.
'''Distributivity'''
For a,b,c ∈ F, a*(b + c) = a*b + a*c.
Sometimes the condition 0 ≠ 1 is also included.
[[Category:Algebra]]
[[Category:Algebra]]