Definition of a Field

A field is a commutative ring in which the set of non-zero elements form a group with respect to multiplication. I.e.,

A set $F$ with two operations $+$ and $\ast$ is called a field if:

• $(F,+)$ is an abelian group,
• $(F^{\ast },\cdot )$ is an abelian group.
• for all $x,y,z\in F$, distributivity holds.