Field of fractions

"Quotient field" redirects here. It should not be confused with a quotient ring.

In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. 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 R is sometimes denoted by Quot(R) or Frac(R).

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. The expression "quotient field" may sometimes run the risk of confusion with the quotient of a ring by an ideal, which is a quite different concept.

A multiplicative identity is not required; this construction can be applied to any non-trivial commutative pseudo-ring with no zero divisors. ^[1]

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 the field of rational functions or field of rational fractions^[2][3][4] and is denoted K(X).

Construction

Let R be any commutative pseudo-ring without zero divisors and at least one nonzero element e. One can construct the field of fractions Quot(R) of 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) if and only if nb=md (we think of the class of (n, d) as the fraction n/d). 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 embedding is given by mapping n to the equivalence class of (en, e). Note that this embedding does not depend on the choice of e. If additionally, R contains a multiplicative identity (that is, R is an integral domain), (en, e) will be equivalent to (n, 1).

The pairs (n, d) from Quot(R) are usually written .

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.

See also

• Total ring of fractions — a generalization of the field of fractions to rings with zero divisors.
• Localization of a ring — which generalizes both the field of fractions and the total ring of fractions.
• Quotient ring — although the quotient rings may be fields, they are entirely distinct from the field of quotients.

References