In abstract algebra, the field of fractions 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 are equivalence classes written as with and in and . The field of fractions of is sometimes denoted by or . Mathematicians refer to this construction as the field of fractions, fraction field, field of quotients, or quotient field. All four are in common usage. 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.
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 , is called the or field of rational fractions and is denoted .
Construction
Let be any integral domain. For with , the fraction denotes the equivalence class of pairs , where is equivalent to if and only if . The field of fractions is defined as the set of all such fractions . The sum of and is defined as , and the product of and is defined as . The embedding of in maps each in to the fraction for any nonzero . This is modelled on the identity . The field of fractions of is characterised by the following universal property: if is an injectivering homomorphism from into a field , then there exists a unique ring homomorphism which 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 which takes every integral domain to its fraction field and every homomorphism to the induced map on fields is the left adjoint of the forgetful functor from the category of fields to . A multiplicative identity is not required for the role of the integral domain; this construction can be applied to any nonzero commutativerng with no nonzero zero divisors. The embedding is given by for any nonzero.
If is the complement of a prime ideal , then is also denoted . When is an integral domain and is the zero ideal, is the field of fractions of .
If is the set of non-zero-divisors in , then is called the total quotient ring. The total quotient ring of an integral domain is its field of fractions, but the total quotient ring is defined for any commutative ring.
Semifield of fractions
The semifield of fractions of an commutative semiring with no zero divisors is the smallest semifield in which it can be embedded. The elements of the semifield of fractions of the commutative semiring are equivalence classes written as with and in .
