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
.
Compare these axioms to the group definition from Section
4.2.1. Note that a field can be considered as two
different kinds of groups, one with respect to multiplication and the
other with respect to addition. Fields additionally require
commutativity; hence, we cannot, for example, build a field from
quaternions. The distributivity axiom appears because there is now an
interaction between two different operations, which was not possible
with groups.
Steven M LaValle
2020-08-14