Field
A field is a structure of abstract algebra, similar to a group or a ring. Informally, fields are the general structure in which the usual laws of arithmetic governing the operations and
hold. In particular, the rational numbers
, the real numbers
, and the complex numbers
are all fields.
Formally, a field is a set of elements with two operations, usually called multiplication and addition (denoted
and
, respectively) which have the following properties:
- A field is a ring. Thus, a field obeys all of the ring axioms.
, where 1 is the multiplicative identity and 0 is the additive indentity. Thus fields have at least 2 elements.
- If we exclude 0, the remaining elements form an abelian group under multiplication. In particular, multiplicative inverses exist for every element other than 0.
Common examples of fields are the rational numbers , the real numbers
, or the integers
taken modulo some prime
, denoted
or
. In each case, addition and multiplication are defined "as usual." Other examples include the set of algebraic numbers and finite fields of order
for
an arbitrary positive integer.