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

Title simple transcendental field extension SimpleTranscendentalFieldExtension 2013-03-22 15:02:20 2013-03-22 15:02:20 pahio (2872) pahio (2872) 11 pahio (2872) Corollary msc 12F99 simple infinite field extension FunctionField