Field of fractions
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In mathematics, every integral domain can be embedded in a field; the smallest field which can be used is the field of fractions 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).
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[edit] Examples
- The field of fractions of the ring of integers is the field of rationals, Q = Quot(Z).
- Let R := { a + bi | a,b in Z } be the ring of Gaussian integers. Then Quot(R) = {c + di | 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).
[edit] Construction
One can construct the field of fractions 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 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.
[edit] 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.
[edit] See also
- Localization of a ring, which generalizes the field of fractions construction
- Quotient ring - although quotient rings may be fields, they are entirely distinct from quotient fields.
