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 :
 * $(1): \quad$ the algebraic structure $\left({F, +}\right)$ is an abelian group
 * $(2): \quad$ the algebraic structure $\left({F^*, \times}\right)$ is an abelian group where $F^* = F \setminus \left\{{0}\right\}$
 * $(3): \quad$ the operation $\times$ distributes over $+$.

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