Definition:Field (Abstract Algebra)/Definition 1

Definition

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

$\struct {F, +, \times}$ is a field if and only if:

$(1): \quad$ 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 $+$.

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, $\mathsf{Pr} \infty \mathsf{fWiki}$ 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.

Examples

Field (Abstract Algebra)/Examples

Also see

• Equivalence of Definitions of Field (Abstract Algebra)

• Results about fields in the context of abstract algebra can be found here.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 87$