Definition:Field (Abstract Algebra)

Definition
A field is a non-trivial division ring whose ring product is commutative.

Thus, let $\left({F, +, \times}\right)$ be an algebraic structure.

Then $\left({F, +, \times}\right)$ is a field iff:
 * the algebraic structure $\left({F, +}\right)$ is an abelian group
 * the algebraic structure $\left({F^*, \times}\right)$ is an abelian group where $F^* = F \setminus \left\{{0}\right\}$
 * the operation $\times$ distributes over $+$.

This definition gives rise to the field axioms, as follows:

Field Axioms
The properties of a field are as follows.

For a given field $\left({F, +, \circ}\right)$, these statements hold true:

These are called the field axioms.