All the textbooks I've seen prove the Strong Nullstellensatz from the Weak Nullstellensatz using the Rabinowitsch trick.
How did they prove the Strong Nullstellensatz before the Rabinowitsch trick was invented?
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$\begingroup$ I guess you mean Hilbert's Nullstellensatz. I tried to find his original proof but could not find it. Don't know how he called the theorem... But I found this: link.springer.com/article/10.1007%2FBF01187946?LI=true#page-1 I did not read it in full because I dont have free access (or at least not from home). But sounds as it takes another path and mentions a proof by Brauer which is it first reference which seems noy to use the Rabinowitsch trick, too. I hope this might help. $\endgroup$ – user103093 Feb 28 '16 at 17:50
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$\begingroup$ Here a link to the paper by Brauer: projecteuclid.org/euclid.bams/1183512373 $\endgroup$ – user103093 Feb 28 '16 at 17:57
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$\begingroup$ Hilbert's original proof is linked to in this answer. $\endgroup$ – Takumi Murayama Mar 1 '16 at 7:39
