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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.
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