My Math Forum Fin. gen field extension -> intermediate field f.g. also?

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 September 18th, 2012, 11:48 AM #1 Senior Member   Joined: Nov 2011 Posts: 100 Thanks: 0 Fin. gen field extension -> intermediate field f.g. also? If $k\subseteq E\subseteq K$ if an extension of fields, with $K$ finitely generated over $k$, how can I go about showing that $E$ must also be finitely generated over $k$? I've tried just some basic degree stuff and looked at examples, but I'm not getting anywhere on this one.. perhaps induct on the number of elements from which $K$ is finitely generated over $k$ that happen to be in $E$? Any help appreciated, thanks!
 September 20th, 2012, 06:53 PM #2 Senior Member   Joined: Aug 2010 Posts: 195 Thanks: 5 Re: Fin. gen field extension -> intermediate field f.g. also I claim that if $K$ is finitely generated over $k$, then there is a natural way to view $K$ as a (finite dimensional!) vector space over $k$. In this way, $E \subseteq K$ can be viewed as a subspace, so what can we say about its dimension?

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