# field

In physics, a field is an entity that is described
by specifying the value of some quantity at every point of space and
time. Alternatively, a region within which a particular type of force can be observed or experienced. Varieties of field include a gravitational
field, electric field, magnetic
field (or when the latter two are linked, an electromagnetic
field), and nuclear field.

The laws of physics suggest that fields represent more than a possibility
of force being observed, but that they can also transmit energy and momentum – a light wave, for example, is a phenomenon completely defined by fields. Whereas
a field exists throughout a region of space and time, a particle exists
only at a single point.

In mathematics, a field is a set *F* (such as a number system) with two operations + and ×, in which
(1) both + and × are associative and commutative, and the operation
× is distributive over +;
(2) there are two identity elements
in *F*, 0 relative to + and 1 relative to ×, such that *a* + 0 = *a* and *a* × 1 = *a* for any element *a* of the field; (3) every element *a* has an inverse
-*a*, also a member of the set, such that *a* + (-*a*)
= 0; (4) every nonzero element *a* has an inverse *a*^{-1},
also a member of the set, such that *a* × *a*^{-1} = 1. Examples of fields include the set of rational
numbers and the real numbers with, in each case, the operations addition and multiplication. A set (with
two operations) which satisfies conditions (1), (2), and (3) but not
(4) (because the result of dividing one integer by another is not necessarily
an integer), and therefore is not a field, is an **integral domain**:
an example is the set of all integers under addition and multiplication. Compare with ring.
See also group.