A "field" is the name given to a pair of numbers and a set of operations which together satisfy several specific laws. A familiar example of a field is the set of rational numbers and the operations addition and multiplication. An example of a set of numbers that is not a field is the set of integers. It is an "integral domain." It is not a field because it lacks multiplicative inverses. Without multiplicative inverses, division may be impossible.

The elements of a field obey the following laws:

1. Closure laws: a + b and ab are unique elements in the field.

2. Commutative laws: a + b = b + a and ab = ba.

3. Associative laws: a + (b + c) = (a + b) + c and a(bc) = (ab)c.

4. Identity laws: there exist elements 0 and 1 such that a + 0 = a and a x 1 = a.

5. Inverse laws: for every a there exists an element -a such that...