simple transcendental field extensionSimpleTranscendentalFieldExtension2005-02-15 16:45:302009-05-18 20:53:19Corollarysimple field extension% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe extension field $K(\alpha)$ of a base field $K$, where $\alpha$ is a transcendental element with respect to $K$, is a \emph{\PMlinkname{simple}{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.
\textbf{Example.}\, The subfields $\mathbb{Q}(\pi)$ and $\mathbb{Q}(e)$ of $\mathbb{R}$, where $\pi$ is \PMlinkname{Ludolph's constant}{Pi} and $e$ Napier's constant, are isomorphic.