field of fractions
The way you build the rationals Q out of the integers Z — by allowing yourself to divide by any nonzero number — works for any integral domain. The field of fractions is exactly this construction: it is the smallest field into which a given integral domain embeds, formed by introducing formal quotients a/b of its elements.
Given an integral domain R, consider pairs (a, b) with a, b in R and b ≠ 0, and declare (a, b) ~ (c, d) when a·d = b·c — the same cross-multiplication rule that says 1/2 = 2/4. The equivalence classes, written a/b, form a field Frac(R) under the usual addition a/b + c/d = (a·d + b·c)/(b·d) and multiplication. The map a ↦ a/1 embeds R into Frac(R), and Frac(R) consists precisely of ratios of elements of R.
The construction has a clean universal property: any injective ring homomorphism from R into a field K factors uniquely through Frac(R). It is the special case of localization where you invert every nonzero element. Integrality is essential — the recipe needs the absence of zero divisors, since otherwise a denominator could multiply something nonzero to zero and the equivalence relation breaks down. For a general commutative ring one instead inverts a multiplicative set and obtains a localization, not a field.
Frac(Z) = Q. For a field k, Frac(k[x]) = k(x), the field of rational functions p(x)/q(x). For the Gaussian integers, Frac(Z[i]) = Q(i).
Three fields of fractions of familiar domains.