Definition:Field of Rational Fractions

Definition
Let $R$ be an integral domain.

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

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

Multiple variables
Let $S$ be a set.

By Polynomial Ring over Integral Domain is Integral Domain, the polynomial ring in $S$ variables $R[\{x_s : s \in S\}]$ over $R$ is an integral domain.

The field of rational fractions in $S$ variables $R(\{x_s : s \in S\})$ is the field of fractions of $R[\{x_s : s \in S\}]$.

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

that can be defined as follows:

Definition 1: by universal property
A field of rational fractions in one variable is a pointed ring extension $(K, \iota, X)$ of $R$ where $K$ is a field, that satisfies the following universal property:
 * For every ring extension $(L, \kappa)$ of $R$ by a field $L$ and for every transcendental element $\alpha \in L$, there exists a unique field homomorphism $f : K \to L$ such that:
 * $f \circ \iota = \kappa$
 * $f(X) = \alpha$

Definition 2: as the quotient ring of a polynomial ring
Let $(R[y], \kappa, y)$ be a polynomial ring over $R$ in one variable $y$.

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

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

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

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