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I have a proof of the primitive element theorem for subfields $K$ of $\mathbb{C}$ which relies on there being infinitely many elements in $K$. In a book I've been reading it states that every finite separable extension is simple and that this follows from the earlier version the primitive element theorem.

Surely this doesn't work if I'm in a finite field? Is there another proof of the primitive element theorem that works in this case? Or is there a more general version that doesn't required infinitely many elements in $K$?

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    @ChrisEagle: oh of course. So separate proofs in each case then.2012-05-17

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A common way of handling this is to deal with the infinite fields using the general theorem, and to deal with the finite fields using the cyclicity of the multiplicative group. A finite degree extension of a finite field is itself finite, and the generator of its multiplicative group then is a primitive element in both senses of the word.