# Quotient field

In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the **quotient field** or the **field of fractions** of the integral domain. The elements of the quotient field of the integral domain *R* have the form *a/b* with *a* and *b* in *R* and *b* ≠ 0. The quotient field of the ring *R* is sometimes denoted by Quot(*R*). The quotient field of the ring of integers is the field of rationals, **Q** = Quot(**Z**). The quotient field of a field is isomorphic to the field itself.

One can construct the quotient field Quot(*R*) of the integral domain *R* as follows: Quot(*R*) is the set of equivalence classes
of pairs *(n, d)*, where *n* and *d* are elements of *R* and *d* is not 0, and the equivalence relation is:
*(n, d)* is equivalent to *(m, b)* iff *nb=md* (we think of the class of *(n, d)* as the fraction *n/d*).
The embedding is given by *n***Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mapsto}**
(*n*, 1). The sum of the equivalence classes of *(n, d)* and *(m, b)* is the class of *(nb + md, db)* and their product is the class of *(mn, db)*.

The quotient field of *R* is characterized by the following universal property: if *f* : *R* → *F* is a ring monomorphism from *R* into a field *F*, then there exists a unique ring monomorphism *g* : Quot(*R*) → *F* which extends *f*.

Assigning to every integral domain its quotient field defines a functor from the category of integral domains (with ring *monomorphisms* as morphisms) to the category of fields. This functor is left adjoint to the forgetful functor which assigns to every field its underlying integral domain.