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. 1.

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

2. 2.

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

3. 3.

   $a+0=a$ for some element $0\in F$ (existence of zero element)

4. 4.

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

5. 5.

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

6. 6.

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

7. 7.

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

8. 8.

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

9. 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 $a\cdot b=1$.