Definition:Field of Rational Functions

Definition

Let $K$ be a field.

Let $K \sqbrk x$ be the integral domain of polynomial forms on $K$.

Let $\map K x$ be the set of rational functions on $K$:

$\map K x := \set {\forall f \in K \sqbrk x, g \in K \sqbrk x^*: \dfrac {\map f x} {\map g x} }$

where $K \sqbrk x^* = K \sqbrk x \setminus \set {\text {the null polynomial} }$.

Then $\map K x$ is the field of rational functions on $K$.