Field of rational functions

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 of the polynomial ring $K[x]$. 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 and of a function field.