# Algebra/Field/Direct/Definition

< Algebra

Field

A set is called a * field* if there are two
binary operations
(called addition and multiplication)

and two different elements , which fulfill the following properties.

- Axioms for the addition:
- Law of associativity: holds for all .
- Law of commutativity: holds for all .
- is the neutral element of the addition, i.e. holds for all .
- Existence of the negative: For every , there exists an element with .

- Axioms of the multiplication:
- Law of associativity: holds for all .
- Law of commutativity: holds for all .
- is the neutral element for the multiplication, i.e. holds for all .
- Existence of the inverse: For every with , there exists an element such that .

- Law of distributivity: holds for all .