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 $F^* = F \setminus \set {0_F}$
 * $(3): \quad$ the operation $\times$ distributes over $+$.

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

Also defined as
Some sources do not insist that the field product of a field is commutative.

That is, what they define as a field, defines as a division ring.

When they wish to refer to a field in which the field product is commutative, the term commutative field is used.