field
A field is a set $F$ together with two binary operations on $F$, called addition and multiplication, and denoted $+$ and $\cdot $, satisfying the following properties, for all $a,b,c\in F$:

1.
$a+(b+c)=(a+b)+c$ (associativity of addition)

2.
$a+b=b+a$ (commutativity of addition)

3.
$a+0=a$ for some element $0\in F$ (existence of zero element^{})

4.
$a+(a)=0$ for some element $a\in F$ (existence of additive inverses)

5.
$a\cdot (b\cdot c)=(a\cdot b)\cdot c$ (associativity of multiplication)

6.
$a\cdot b=b\cdot a$ (commutativity of multiplication)

7.
$a\cdot 1=a$ for some element $1\in F$, with $1\ne 0$ (existence of unity element)

8.
If $a\ne 0$, then $a\cdot {a}^{1}=1$ for some element ${a}^{1}\in F$ (existence of multiplicative inverses^{})

9.
$a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ (distributive property)
Equivalently, a field is a commutative ring $F$ with identity^{} such that:

•
$1\ne 0$

•
If $a\in F$, and $a\ne 0$, then there exists $b\in F$ with $a\cdot b=1$.
Title  field 

Canonical name  Field 
Date of creation  20130322 11:48:43 
Last modified on  20130322 11:48:43 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  9 
Author  djao (24) 
Entry type  Definition 
Classification  msc 12E99 
Classification  msc 03A05 