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Does anyone know Hilbert's original proof of his basis theorem--the non-constructive version that caused all the controversy? I know this was circa 1890, and he would have proved it for $\mathbb{C}[x_1,\ldots,x_n]$. Since this was before Emmy Noether's "Ideal theory for Rings", he would not have had the ascending chain condition to prove it. I wanted to know how his original proof compares with Emmy's a decade or so later, and in particular how his proof hints at the formulation of the ascending chain condition.

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You can find Hilbert's "Gesammelte Abhandlungen" (collected publications) in digitized form at the University of Goettingen: The documents are in German. – Axel Kemper Apr 7 '13 at 18:08
A related survey with lots of references:… – Axel Kemper Apr 7 '13 at 18:27
Thanks! I'll take a look at those! :D – Lucy Apr 8 '13 at 0:07

If you can get a hold of: "Theory Of Algebraic Invariants" by David Hilbert (pp.126-130) , Lecture XXXV ( July 9, 1897) the original proof is given. This book is an English translation published by Cambridge University Press.

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