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Encyclopedia > Field of fractions

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). Euclid, detail from The School of Athens by Raphael. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... This article presents the essential definitions. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

Contents


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)={a+bi | a,b 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).

The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ... In mathematics, the field of Gaussian rationals is the field Q(i) formed by adjoining the imaginary number i to the field of rationals. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ...

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 nmapsto(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). In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...


The field of fractions of R is characterized by the following universal property: if f : RF 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 under certain conditions. ... In 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. 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 mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... A forgetful functor is a type of functor in mathematics. ...


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. Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ...


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.

  Results from FactBites:
 
Fraction (289 words)
A fraction consisting of two integers is called a rational number.
If both the numerator and the denominator of a fraction are multiplied or divided by the same number, then the fraction does not change its value.
To divide one fraction by another one, flip numerator and denominator of the second one, and then multiply the two fractions.
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