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, ${\mathbb{Q}}$; 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.