- Field of fractions
In

mathematics , everyintegral domain can be embedded in a field; the smallest field which can be used is the**field of fractions**or**field of quotients**of the integral domain. The elements of the field of fractions of the integral domain "R" have the form "a/b" with "a" and "b" in "R" and "b" ≠ 0. The field of fractions of the ring "R" is sometimes denoted by Quot("R") or Frac("R").**Examples*** The field of fractions of the ring of

integer s is the field of rationals,**Q**= Quot(**Z**).

* Let "R" := { "a" + "b" i | "a","b" in**Z**} be the ring ofGaussian integer s. Then Quot("R") = {"c" + "d" i | "c","d" in**Q**}, the field ofGaussian rational s.

* The field of fractions of a field is isomorphic to the field itself.

* Given a field "K", the field of fractions of thepolynomial ring in one indeterminate "K" ["X"] (which is an integral domain), is called**field of rational functions**and denoted "K"("X").**Construction**One can construct the field of fractions Quot("R") of the integral domain "R" as follows: Quot("R") is the set of

equivalence class esof pairs "(n, d)", where "n" and "d" are elements of "R" and "d" is not 0, and theequivalence 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" $mapsto$("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 field of fractions of "R" is characterized by the following

universal property : if "f" : "R" → "F" is an injectivering homomorphism from "R" into a field "F", then there exists a unique ring homomorphism "g" : Quot("R") → "F" which extends "f".There is a categorical interpretation of this construction. Let

**C**be the category of integral domains andinjective ring maps. Thefunctor from**C**to the category of fields which takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the left adjoint of theforgetful functor from the category of fields to**C**.**Terminology**Mathematicians refer to this construction as the "quotient field", "field of fractions", or "fraction field". All three are in common usage, and which is used is a matter of personal taste. Those who favor the latter two sometimes claim that the name quotient field incorrectly suggests that the construction is related to taking a quotient of the ring by an ideal.

**See also***

Localization of a ring , which generalizes the field of fractions construction

*Quotient ring - although the quotient rings may be fields, they are entirely distinct from the field of quotients.

*Total ring of fractions - a generalization of the field of fractions

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