In
abstract algebra, the field of fractions of an
integral domain is the smallest
field in which it can be
embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of
integers and the field of
rational numbers. Intuitively, it consists of ratios between integral domain elements.
The field of fractions of an integral domain is sometimes denoted by or , and the construction is sometimes also called the fraction field, field of quotients, or quotient field of . All four are in common usage, but are not to be confused with the
quotient of a ring by an ideal, which is a quite different concept. For a
commutative ring that is not an integral domain, the analogous construction is called the
localization or ring of quotients.
Definition
Given an integral domain and letting , we define an
equivalence relation on by letting whenever . We denote the
equivalence class of by . This notion of equivalence is motivated by the rational numbers , which have the same property with respect to the underlying
ring of integers.
Then the field of fractions is the set with addition given by
and multiplication given by
One may check that these operations are well-defined and that, for any integral domain , is indeed a field. In particular, for , the multiplicative inverse of is as expected: .
The embedding of in maps each in to the fraction for any nonzero (the equivalence class is independent of the choice ). This is modeled on the identity .
The field of fractions of is characterized by the following
universal property:
if is an
injectivering homomorphism from into a field , then there exists a unique ring homomorphism that extends .
There is a
categorical interpretation of this construction. Let be the
category of integral domains and injective ring maps. The
functor from to the
category of fields that 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
inclusion functor from the category of fields to . Thus the category of fields (which is a full subcategory) is a
reflective subcategory of .
A
multiplicative identity is not required for the role of the integral domain; this construction can be applied to any
nonzero commutative
rng with no nonzero
zero divisors. The embedding is given by for any nonzero .[1]
Examples
The field of fractions of the ring of
integers is the field of
rationals: .
The field of fractions of a field is canonically
isomorphic to the field itself.
Given a field , the field of fractions of the
polynomial ring in one indeterminate (which is an integral domain), is called the field of rational functions, field of rational fractions, or field of rational expressions[2][3][4][5] and is denoted .