Rational functions/Field/2/Introduction/Section

The polynomial ring ${}K[X]$ is a commutative ring, but not a field. However, we can construct a field which contains the polynomial ring with the help of the so-called formal-rational functions, in a similar way as we can construct the rational numbers ${}\mathbb {Q}$ from the integers ${}\mathbb {Z}$. For this, we define

${}K(X):={\left\{{\frac {P}{Q}}\mid P,Q\in K[X],\,Q\neq 0\right\}}\,,$

where we identify, like in ${}\mathbb {Q}$, two fractions ${}{\frac {P}{Q}}$ and ${}{\frac {P'}{Q'}}$, whenever

${}PQ'=P'Q\,$

holds. In this way, the field of rational functions (over ${}K$) arises.

A fraction ${}P/Q$ of polynomials may be considered as a function which is defined outside the zeroes of the denominator. The example shows the graph of the rationale function ${}1/X$.

The formal expression ${}P/Q$ can be considered as a function in the following way.

Definition

Let ${}K$ be a field. For polynomials ${}P,Q\in K[X]$, ${}Q\neq 0$, the function

$D\longrightarrow K,z\longmapsto {\frac {P(z)}{Q(z)}},$

where ${}D$ is the complement of the zeroes

of ${}Q$, is called a rational function.

Next to the polynomial functions, the simplest functions are the rational functions.