# Definition:Field of Rational Fractions

Not to be confused with Definition:Field of Rational Functions or Definition:Field of Rational Numbers.

## Definition

Let $R$ be an integral domain.

## As a ring

### One variable

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

The field of rational fractions in one variable $\map R x$ is the field of fractions of $R \sqbrk x$.

### Multiple variables

Let $S$ be a set.

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

The field of rational fractions in $S$ variables $\map R {\set {x_s : s \in S} }$ is the field of fractions of $R \sqbrk {\set{ x_s : s \in S} }$.

## As an algebra

### One variable

A field of rational fractions in one variable is an ordered triple $\struct {K, \iota, x}$ 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 $\struct {K, \iota, X}$ 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:
$f \circ \iota = \kappa$
$\map f X = \alpha$

#### Definition 2: as the quotient ring of a polynomial ring

Let $\struct {R \sqbrk y, \kappa, y}$ be a polynomial ring over $R$ in one variable $y$.

By Polynomial Ring over Integral Domain is Integral Domain, $R \sqbrk y$ is an integral domain.

Let $\struct {K, \lambda}$ be its field of fractions.

Then the field of rational fractions is the ordered triple $\struct {K, \lambda \circ \kappa, \map \lambda y}$.

## 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.