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;
 * 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.