#### Fields

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:

1. (Associativity) For all , and .
2. (Commutativity) For all , and .
3. (Distributivity) For all , .
4. (Identities) There exist , such that for all .
5. (Additive Inverses) For every , there exists some such that .
6. (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