Suppose $L,K$ are fields. Is is true that if $L$ a finitely generated $K$-algebra then $L/K$ is a finite field extension?

Wikipedia seems to think so. But if it is true surely it's difficult to prove? After all the Nullstellensatz would seem to follow immediately from such a result. Is this the basic idea of Noether Normalisation?

Any advice or guidance about a proof would be greatly appreciated!

  • 1
    $\begingroup$ @EdwardHughes: This is one form of Nullstellensatz. It is not easier nor harder than Nullstellensatz. $\endgroup$ – user18119 Oct 7 '12 at 14:03

You can look at my answer given here. It proves using Noether Normalisation that the extension $L/K$ must necessarily be finite.

  • $\begingroup$ Aha - thank you. Didn't see that question. I guess mine is a slight duplicate of that one. As such it can probably just be closed! $\endgroup$ – Edward Hughes Oct 7 '12 at 14:09

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