Polynomials are usually defined over a *field*, which is another
object from algebra. A field is similar to a group, but it has more
operations and axioms. The definition is given below, and while
reading it, keep in mind several familiar examples of fields: the
rationals,
; the reals,
; and the complex plane,
. You
may verify that these fields satisfy the following six axioms.

A *field* is a set
that has two binary operations,
(called *multiplication*) and
(called *addition*), for which the
following axioms are satisfied:

- (
**Associativity**) For all , and . - (
**Commutativity**) For all , and . - (
**Distributivity**) For all , . - (
**Identities**) There exist , such that for all . - (
**Additive Inverses**) For every , there exists some such that . - (
**Multiplicative Inverses**) For every , except , there exists some such that .

Steven M LaValle 2020-08-14