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.
- 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.