Field of fractions

Definition

Let $R$ be an integral domain. On the set of pairs $(a, b)$ with $a, b \in R$ and $b \neq 0$, let $(a, b) \sim (c, d)$ if $a d = b c$. This is an equivalence relation. Write $a/b$ for the class of $(a, b)$. The field of fractions $\operatorname{Frac}(R)$ is the set of these classes, with the operations $$\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}, \qquad \frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}.$$ It is a field, and the map $R \to \operatorname{Frac}(R)$, $a \mapsto a/1$, is an injective ring homomorphism.

Remarks

The product $b d$ is not zero because $R$ is an integral domain. This makes the operations well defined. The inverse of a nonzero $a/b$ is $b/a$. Universal property: every injective ring homomorphism from $R$ to a field $K$ extends to exactly one ring homomorphism from $\operatorname{Frac}(R)$ to $K$. The field of fractions of the polynomial ring $K[x]$ over a field $K$ is the field $K(x)$ of rational functions.

Explanations

L1 human

This is how the rational numbers come from the integers. A fraction is a pair of a numerator and a nonzero denominator. Two fractions are equal when cross-multiplication gives the same result, as with $1/2 = 2/4$.