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 where $F^* = F \setminus \left\{{0}\right\}$
 * the operation $\times$ distributes over $+$.

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

Linguistic Note
Note that while in English the word for this entity is field, its name in other European languages translates as body.

The original work was done by, who used the word Körper. When translating his work into English, mistakenly used the word field. The translator into French did not make the same mistake.