Field of fractions - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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). In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

1 Examples
2 Construction
3 Terminology
4 See also

Examples

The field of fractions of the ring of integers is the field of rationals, Q = Quot(Z). The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions
Let R := { a + b i | a,b in Z } be the ring of Gaussian integers. A Gaussian integer is a Complex number whose real and imaginary part are both Integers The Gaussian integers with ordinary addition and multiplication of complex Then Quot(R) = {c + d i | c,d in Q}, the field of Gaussian rationals. In Mathematics, a Gaussian rational number is a Complex number of the form p + qi, where p and q are both
The field of fractions of a field is isomorphic to the field itself. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective
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). In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables

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). In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" ↔ 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. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication

There is a categorical interpretation of this construction. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, in the area of Category theory, a forgetful functor is a type of Functor.

Terminology

Mathematicians refer to this construction as the quotient field, field of fractions, or fraction field. A mathematician is a person whose primary area of study and research is the field of Mathematics. 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.

Contents

Examples

Construction

Terminology

See also