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.
Referenced By
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