## Rational Numbers and Fields

This chapter introduces as new elements certain couples (m, n) of positive integers, where each couple is to behave as if it were the solution of the equation n + x = m. Both the integral domain Q of all rational numbers and the integral domain R of all real numbers have an essential algebraic advantage over the domain Z of integers. Commutative rings with this property are called fields; the chapter shows that division is possible and has its familiar properties in any commutative ring where all nonzero elements have nonmultiplicative inverses. It shows that any integral domain can be extended to a field in one and only one minimal way. The method of extension is illustrated by the standard representation of fractions as quotients of integers. All the rules for algebraic manipulation are satisfied in fields, considered as integral domains. The usual rules for the manipulation of quotients can also be proved from the postulates for a field.