simple transcendental field extension

The extension field $K(\alpha )$ of a base field $K$, where $\alpha$ is a transcendental element with respect to $K$, is a simple (http://planetmath.org/SimpleFieldExtension) transcendental extension of $K$.  All such extension fields are isomorphic to the field $K(X)$ of rational functions in one indeterminate $X$ over $K$, and thus to each other.

Example.  The subfields $\mathbb{Q}(\pi )$ and $\mathbb{Q}(e)$ of $ℝ$, where $\pi$ is Ludolph’s constant (http://planetmath.org/Pi) and $e$ Napier’s constant, are isomorphic.