- For the field in relations, see field (relation).
A field is a set paired with two operations on the set, which are designated as addition and multiplication . As a group can be conceptualized as an ordered pair of a set and an operation, , a field can be conceptualized as an ordered triple .
A set with addition and multiplication, , is a field if and only if it satisfies the following properties:
- Commutativity of both addition and multiplication — For all , and
- Associativity of both addition and multiplication — For all , and
- Additive Identity — There exists a "zero" element, , called an additive identity, such that for all
- Additive Inverses — For each , there exists a , called an additive inverse of , such that
- Multiplicative Identity — There exists a "one" element, , different from 0, called a multiplicative identity, such that for all
- Multiplicative Inverses — For each , except for 0, there exists a , called a multiplicative inverse of , such that
- Distributive property — For all ,
- Closure of addition and multiplication — For all , and
Alternatively, a field can be defined as a commutative ring with unity (has a multiplicative identity) and multiplicative inverses.
We will often abbreviate the multiplication of two elements, , by juxtaposition of the elements, . Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, .
We can also denote and as additive and multiplicative inverses of any . Furthermore, we can define two more operations, called subtraction and division by , and provided that , .
Important Results
Because a field is also a ring with unity, these properties are inherited:
- is an abelian groups
- , for all
- , for all
- , for all
- , for all
- Multiplication distributes over subtraction.
Additionally:
- is also an abelian group, where is the set of nonzero elements of
- Any field contains a subfield that is field-isomorphic to or for some prime .
Optional Properties
A field is:
- A subfield of a field if (see subset), where addition and multiplication on is a domain restriction on the addition and multiplication on . More commonly, we say that is an extension field of , and in fact, is also a vector space over
- An ordered field if there exists a total order on such that for all , if , then , (translation invariance), and if and , then
Examples
- Under the usual operations of addition and multiplication, the rational numbers (), algebraic numbers (), real numbers (), and complex numbers () are fields.
- An extension field of , such as .
Related
Elements of a field are the quantities over the vectorspaces are constructed and there are also called the scalars.
In the same branch functions , where is a field are called scalar fields.