Field of fractions

The rationals are built from the integers by forming pairs \((a,b)\) with \(b\neq0\) and declaring that \((a,b)\) and \((c,d)\) represent the same fraction when \(ad=bc\). The same recipe works in any integral domain \(D\): the classes of pairs form the field \(\operatorname{Frac}(D)\), with the usual addition and multiplication. The absence of zero divisors in \(D\) is what makes the relation transitive and keeps the product of two nonzero denominators nonzero.

The most common examples are \(\operatorname{Frac}(\mathbb{Z})=\mathbb{Q}\), the field of rational functions \(\operatorname{Frac}(F[x])=F(x)\) over a field \(F\), and \(\operatorname{Frac}(\mathbb{Z}[i])=\mathbb{Q}(i)\). The domain \(D\) is identified with the fractions \(a/1\), so \(D\) sits inside its field of fractions, and if \(D\) is already a field nothing is added.

It is the smallest field containing \(D\): any field \(K\) containing \(D\) contains the quotients \(ab^{-1}\), that is, a copy of \(\operatorname{Frac}(D)\). This universal property is what is used when a result about polynomials is proved by passing to rational functions, for example to factor in \(\mathbb{Z}[x]\) using \(\mathbb{Q}[x]\).