the field extension $\mathbb{R}/\mathbb{Q}$ is not finiteExtensionMathbbRmathbbQIsNotFinite2003-09-11 17:26:592005-05-02 16:46:08Corollarya finite extension of fields is an algebraic extensionpitranscendentalrealsrationals% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}\begin{thm}
Let $L/K$ be a finite field extension. Then $L/K$ is an algebraic
extension.
\end{thm}
\begin{cor}
The extension of fields $\Reals/\Rats$ is not finite.
\end{cor}
\begin{proof}[Proof of the Corollary]
If the extension was finite, it would be an algebraic extension. However, the
extension $\Reals/\Rats$ is clearly not algebraic. For example,
$e\in\Reals$ is transcendental over $\Rats$ (see e is transcendental).
\end{proof}