Field

From AoPSWiki

A field is a structure of abstract algebra, similar to a group or a ring. A field is a set of elements with two operations, usually called multiplication and addition and denoted and , 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 the operation . In particular, multiplicitive 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."