---
tag: 4V81611P9N
kind: concept
name: Field of rational functions
msc: [12F20, 13B30]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_function
---

## Summary

The field of rational functions $K(x)$ over a field $K$ consists of the quotients $p/q$ of polynomials with $q \ne 0$.
It is the [[field of fractions|0F2PTGX1A2]] of the [[polynomial ring|0SDC934R52]] $K[x]$.
It is the simplest field that contains $K$ and an element $x$ that is not a root of any nonzero polynomial over $K$.
It is the basic example of a [[purely transcendental extension|G5PVJMM37Z]] and of a [[function field|XKEXQDRTGH]].

## Explanations

### L1 human

Polynomials can be added, subtracted and multiplied, but not always divided.
Allowing fractions of polynomials, as with fractions of integers, gives a field.
For example, $\frac{1}{x}$ and $\frac{x^2 + 1}{x - 1}$ lie in $\Q(x)$.
Two fractions are equal when cross multiplication gives the same polynomial, as for $\frac{x}{x^2} = \frac{1}{x}$.
The [[definition|JX6CHG7HWV]] makes this precise.