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 $K$:

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

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

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

Also see

• Field of Rational Functions is Field

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Example $26$