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. The quotient field of a field is that 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 |-> (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 homomorphism from *R* into a field *F*, then there exists a unique ring homomorphism *g* : Quot(*R*) `->` *F* which extends *f*.

Assigning to every integral domain its quotient field defines a functor from the category of integral domains to the category of fields. This functor is left adjoint to the forgetful functor which assigns to every field its underlying integral domain.