Fields

Definition.

A (commutative) field is a set $K$ with two internal laws $+$ (addition) and $\times$ (multiplication) and two distinct elements $0$ and $1$ satisfying the following properties:

1. $(K,+,0)$ is a commutative (abelian) group
2. $(K\setminus \{0\},\times ,1)$ is a commutative group
3. the law $\times$ is distributive with respect to the law $+$: for any $a$, $b$, and $c$ in $K$, $a\times (b+c)=a\times b+a\times c$.