Definition:Field (Abstract Algebra)

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

Thus, let $\struct {F, +, \times}$ be an algebraic structure.

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

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