Definition:Field of Rational Fractions
- Not to be confused with Definition:Field of Rational Functions or Definition:Field of Rational Numbers.
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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 $\struct {K, \lambda \circ \kappa, \map \lambda y}$.
Multiple variables
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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.