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Weak nullstellensatz describes maximal ideals in polynomial rings over algebraically closed fields at least when the cardinality number of variables is finite. Lang obtained the same conclusion also when the transcendence degree of the field over its prime field exceeds the number of variables (I don't know if "weak nullstellensatz" officially now includes Lang's extension, but for here let's say it does.)

How explicitly can we describe the maximal spectrum of polynomial rings over algebraically closed fields when weak nullstellensatz fails?

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@David: I am pretty sure that "weak Nullstellensatz" does not include the extension by Lang of which you speak. In fact, I know the former pretty well (my notes on commutative algebra contain something like five proofs of it), but I think I have never heard of this result of Lang. It certainly sounds nice: could you give a reference? –  Pete L. Clark Feb 1 '11 at 6:29
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Never mind: it was easy enough to find: Lang, Serge Hilbert's Nullstellensatz in infinite-dimensional space. Proc. Amer. Math. Soc. 3, (1952). 407–410. I'll certainly take a look. –  Pete L. Clark Feb 1 '11 at 6:35
    
Pete - I'm happy to have brought Lang's article to your attention! It always interests me how results/proofs do or don't get into the canon. For example, the American Mathematical Monthly has an endless supply of improvements on basic textbook proofs, but most seem ignored as new textbooks copy proofs out of old... –  David Feldman Feb 1 '11 at 6:51
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An explicit description of all maximal ideals seems unlikely to me in the general case. Lang's article gives an example where the residue field is isomorphic to $k(t)$, and this example can probably be adapted to find arbitrary residue fields $K/k$ with $\operatorname{trdeg}(K/k) \leq$ the number of variables. –  François Brunault Feb 1 '11 at 14:29
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@François If I may make a comparison, an explicit description of even a single non-principal ultrafilter on $\Bbb N$ more than seems unlikely - such a description would surely embody a proof of an axiom, BPIT, provably independent of $ZF$ . But that has not prevented the development of a whole literature concerning the structure of $\beta{\Bbb N}$ (including more independence results). So my question which asks for a description of the maximal spectrum need not fall to the difficulty of describing the individual ideals. –  David Feldman Feb 2 '11 at 6:04
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