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September 18th, 2012, 12:48 PM   #1
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Fin. gen field extension -> intermediate field f.g. also?

If if an extension of fields, with finitely generated over , how can I go about showing that must also be finitely generated over ?

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 is finitely generated over that happen to be in ?

Any help appreciated, thanks!
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September 20th, 2012, 07:53 PM   #2
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Re: Fin. gen field extension -> intermediate field f.g. also

I claim that if is finitely generated over , then there is a natural way to view as a (finite dimensional!) vector space over . In this way, can be viewed as a subspace, so what can we say about its dimension?
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