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
.