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 $1 \neq 0$ (existence of unity element)
8. If $a \neq 0$ , 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:

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