I'm reading the following article by Maxwell Rosenlicht:(adsbygoogle = window.adsbygoogle || []).push({});

http://www.jstor.org/stable/2318066

(The question should be clear without the article, but I present it here for reference.)

In the beginning of the article he discusses differential fields (i.e. a field [itex]F[/itex] with a map [itex]F\to F[/itex], [itex]a\mapsto a'[/itex] such that [itex](a+b)'=a'+b'[/itex] and [itex](ab)'=a'b+ab'[/itex]). He presents the result that given a differential field [itex]F[/itex] and a separable extension [itex]K/F[/itex], there exists a unique differential structure on [itex]K[/itex] that extends that on [itex]F[/itex]. After showing uniqueness quite easily, he proceeds to show existence. His first sentence toward showing existence is, "Using the usual field-theoretic arguments, we may assume that [itex]K[/itex] is a finite extension of [itex]F[/itex], so that we can write [itex]K=F(x)[/itex], for a certain [itex]x\in K[/itex]."

Of course if the extension is finite, then it is simple, so I understand the second part of the sentence. But what are the "usual field-theoretic arguments" he's referring to? It's not clear to me why we can assume that this extension is finite.

If this is too complicated to explain quickly, perhaps you could give me a reference in Dummit and Foote or an internet link.

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# For separable extensions, why may we argue as if they're finite?

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