Definition:Field of Rational Fractions


 * Not to be confused with Definition:Field of Rational Functions

In one indeterminate
Let $R$ be an integral domain.

Condensed definition
By Polynomial Ring over Integral Domain is Integral Domain, the polynomial ring in one indeterminate $R[x]$ over $R$ is an integral domain.

The field of rational fractions in one indeterminate $R(x)$ is the field of fractions of $R[x]$.

Full definition
A field of rational fractions in one indeterminate is an ordered triple $(K, f, x)$ where:
 * $K$ is a field
 * $f : R \to K$ is a unital ring homomorphism, called canonical embedding
 * $x$ is an element of $K$, called indeterminate

that can be defined as follows:

Let $(R[y], g, y)$ be a polynomial ring over $R$ in one indeterminate $y$.

By Polynomial Ring over Integral Domain is Integral Domain, $R[y]$ is an integral domain.

Let $(K, \iota)$ be its field of fractions.

Then the field of rational fractions is the ordered triple $(K, \iota \circ g, \iota(y))$.

Also known as
The field of rational fractions is commonly referred to as the field of rational functions. Strictly speaking, field of rational functions has a different meaning.

Also see

 * Universal Property of Field of Rational Fractions
 * Definition:Rational Function
 * Definition:Polynomial Ring
 * Definition:Field of Rational Functions