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Quotient field

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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