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. In each case, addition and multiplication are "as usual." Other examples include the set of algebraic numbers and finite fields other than the integers modulo a prime.