# Field Theory/Definition, elementaries

Definition (Field):

A field is a set ${\displaystyle \mathbb {F} }$with two operations * and + that satisfy the following axioms:

1. ${\displaystyle (a+b)+c=a+(b+c)}$
2. ${\displaystyle a+b=b+a}$
3. ${\displaystyle (a*b)*c=a*(b*c)}$
4. ${\displaystyle a*b=b*a}$
5. ${\displaystyle a+0=a}$
6. ${\displaystyle a*1=a}$
7. ${\displaystyle a+(-a)=0}$
8. ${\displaystyle a*a^{-1}=1}$
9. ${\displaystyle a*(b+c)=a*b+a*c}$