## Field Axioms

A field is a setthat is a commutative group with respect to two compatible operations, addition and multiplication, with "compatible" being formalized by distributivity, and the caveat that the additive identity (0) has no multiplicative inverse (one cannot divide by 0).

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted byandrespectively, such that the following axioms hold;

- Closure ofunder addition and multiplication
- For all (or more formally,andare binary operations on).
- Associativity of addition and multiplication
- For alland
- Commutativity of addition and multiplication
- For all
- Additive and multiplicative identity
- There exists an element ofcalled the additive identity element and denoted by 0, such that for allLikewise, there is an element, called the multiplicative identity element and denoted by 1, such that for allThe additive identity and the multiplicative identity may not be the same.
- Additive and multiplicative inverses
- For everythere existssuch thatSimilarly, for any other than 0, there exists an elementsuch that(The elements andare denotedandrespectively.) In other words subtraction and division operations exist.
- Distributivity of multiplication over addition
- For all

Note that all but the last axiom are exactly the axioms for a commutative group, while the last axiom is a compatibility condition between the two operations.

Examples:are all fields as isfor n prime. The tables below are for a finite field with four elements.

0 | 1 | A | B | 0 | 1 | A | B | |||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | A | B | |

1 | 0 | 1 | A | B | 1 | 1 | 0 | B | A | |

A | 0 | A | B | 1 | A | A | B | 0 | 1 | |

B | 0 | B | A | 1 | B | B | A | 1 | 0 |