Field Theory/Definition, elementaries

Definition (Field):

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

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