field

A field is an algebraic structure consisting of commutative ring, with identities $0\ne 1$, and where all nonzero elements are invertible.

Formally,
A field is a set $F$ equipped with two binary operations, addition and multiplication, such that for all elements $a,b,c\in F$,

$(F,+,\times )$ is a ring:

1. $(F,+)$ is closed: $a+b\in F$
2. $(F,+)$ is associative: $(a+b)+c=a+(b+c)$
3. $(F,+)$ has an identity element: $a+0=a=0+a$
4. $(F,+)$ has inverses: $a+(-a)=0=(-a)+a$
5. $(F,+)$ is commutative: $a+b=b+a$
6. $(F,\times )$ is closed: $ab\in F$
7. $(F,\times )$ is associative: $(ab)c=a(bc)$
8. $(F,\times )$ has an identity element: $a1=a=1a$
9. $(F,+,\times )$ is left-distributive: $a(b+c)=ab+ac$
10. $(F,+,\times )$ is right-distributive: $(a+b)c=ac+bc$

Additionally, $(R,\times )$ is an abelian group:

11. $(F,\times )$ is commutative: $ab=ba$
12. Identities are unique: $0\ne 1$
13. $(F,\times )$ has inverses: $a(a^{-1})=1=(a^{-1})a$