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, $ {\mathbb{Q}}$; the reals, $ {\mathbb{R}}$; and the complex plane, $ {\mathbb{C}}$. You may verify that these fields satisfy the following six axioms.

A field is a set $ {\mathbb{F}}$ that has two binary operations, $ \cdot : {\mathbb{F}}
\times {\mathbb{F}}\rightarrow {\mathbb{F}}$ (called multiplication) and $ + : {\mathbb{F}}
\times {\mathbb{F}}\rightarrow {\mathbb{F}}$ (called addition), for which the following axioms are satisfied:

  1. (Associativity) For all $ a,b,c \in {\mathbb{F}}$, $ (a+b)+c = a+(b+c)$ and $ (a \cdot b) \cdot c = a \cdot (b \cdot c)$.
  2. (Commutativity) For all $ a,b \in {\mathbb{F}}$, $ a + b = b + a$ and $ a \cdot b = b \cdot a$.
  3. (Distributivity) For all $ a,b,c \in {\mathbb{F}}$, $ a \cdot (b + c)
= a \cdot b + a \cdot c$.
  4. (Identities) There exist $ 0, 1 \in {\mathbb{F}}$, such that $ a + 0 =
a \cdot 1 = a$ for all $ a \in {\mathbb{F}}$.
  5. (Additive Inverses) For every $ a \in {\mathbb{F}}$, there exists some $ b \in {\mathbb{F}}$ such that $ a + b = 0$.
  6. (Multiplicative Inverses) For every $ a \in F$, except $ a = 0$, there exists some $ c \in {\mathbb{F}}$ such that $ a \cdot c = 1$.
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 2012-04-20