A field is a set
that 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 by
and
respectively, such that the following axioms hold;
- Closure ofunder addition and multiplication
- For all
(or more formally,and
are binary operations on). - Associativity of addition and multiplication
- For all
and
- Commutativity of addition and multiplication
- For all
- Additive and multiplicative identity
- There exists an element of
called the additive identity element and denoted by 0, such that for all
Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all
The additive identity and the multiplicative identity may not be the same. - Additive and multiplicative inverses
- For every
there exists
such that
Similarly, for any
other than 0, there exists an element
such that
(The elements
and
are denoted
and
respectively.) 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 is
for 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 |
