[[Polynomial ring]]
# Field of rational functions

Let $D$ be an [[integral domain]] and $D[x]$ be the [[polynomial ring]] over $D$ in indeterminate $x$,
which [[The polynomial ring over an integral domain is an integral domain|is itself an integral domain]]
The **field of rational functions** $D(x)$ in indeterminate $x$ consists of ratios of polynomials in indeterminate $x$ #m/def/ring
$$
\begin{align*}
f(x) = \frac{p(x)}{q(x)}
\end{align*}
$$
where $p(x),q(x) \in D[x]$,
and is the [[field of fractions]]
$$
\begin{align*}
D(x) = \Frac(D[x]) = \Frac(\Frac(D)[x])
\end{align*}
$$
and a [[division algebra]] over $\Frac(D)$.

> [!missing]- Proof
> #missing/proof

## Properties

- [[Lower bound on the dimension of the field of rational functions]]

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