A field is a structure in 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, although there are many others, including subfields of those fields.
Formally, a field (here the letter stands for Körper, the German word for a mathematical field) is a set of elements with two operations, usually called multiplication and addition (denoted and , respectively) which have the following properties:
- is an Abelian group with an identity of .
- (also denoted as ) is also an Abelian group with an identity of .
- Multiplication () distributes over addition (); for any ,
There is also a unique name for , which most accept as the group of units of . Furthermore, it can be proven that the group of units of is a cyclic group for any field which can help in determining certain homomorphisms between fields.
Common examples of fields are the rational numbers , the real numbers , or (the integers modulo for some prime ). In general, a field of order is denoted as , although this is rather unspecific since fields are usually referenced by name.