Field of fractions


Field of fractions

In mathematics, every integral 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 integers is the field of rationals, Q = Quot(Z).
* Let "R" := { "a" + "b" i | "a","b" in Z } be the ring of Gaussian integers. Then Quot("R") = {"c" + "d" i | "c","d" in Q}, the field of Gaussian rationals.
* The field of fractions of a field is isomorphic to the field itself.
* Given a field "K", the field of fractions of the polynomial 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 classesof 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" 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 injective ring 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 and injective ring maps. The functor 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 the forgetful 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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Field (mathematics) — This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation). In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it …   Wikipedia

  • Field extension — In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties. For… …   Wikipedia

  • field of quotients — Math. a field whose elements are pairs of elements of a given commutative integral domain such that the second element of each pair is not zero. The field of rational numbers is the field of quotients of the integral domain of integers. Also… …   Universalium

  • field of quotients — Math. a field whose elements are pairs of elements of a given commutative integral domain such that the second element of each pair is not zero. The field of rational numbers is the field of quotients of the integral domain of integers. Also… …   Useful english dictionary

  • Field desorption — [ mass spectrometer at right] Field desorption (FD)/field ionization (FI) refers to an ion source for mass spectrometry first reported by Beckey in 1969. [Beckey H.D. Field ionization mass spectrometry. Research/Development, 1969 , 20(11), 26] In …   Wikipedia

  • Algebraic number field — In mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector… …   Wikipedia

  • Function field of an algebraic variety — In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V . In complex algebraic geometry these are meromorphic functions and their higher dimensional analogues; in… …   Wikipedia

  • Function field (scheme theory) — In algebraic geometry, the function field KX of a scheme X is a generalization of the notion of a sheaf of rational functions on a variety. In the case of varieties, such a sheaf associates to each open set U the ring of all rational functions on …   Wikipedia

  • Local field — In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non discrete topology.[1] Given such a field, an absolute value can be defined on it. There are two basic types of local field …   Wikipedia

  • Global field — In mathematics, the term global field refers to either of the following:*a number field, i.e., a finite extension of Q or *the function field of an algebraic curve over a finite field, i.e., a finitely generated field of characteristic p >0 of… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.