---- > [!theorem] Theorem. ([[Nullstellensatz]]) > "Hilbert's Nullstellensatz" refers to a set of closely related results. Here are a few — each was used to prove its successor. > 1. [[Zariski's Lemma]] > 2. [[weak Nullstellensatz]] > 3. [[strong Nullstellensatz]] > 4. [[Hilbert's geometry-algebra correspondence]] > 5. Corollary of (4): [[Hilbert's geometry-algebra correspondence|correspondence between minimal nonempty algebraic sets and maximal ideals]]. > 6. Corollary of (5): for $k=\overline{k}$ [[algebraically closed]], the [[maximal ideal|maximal ideals]] of $k[T_{1},\dots,T_{n}]$ are precisely of the form $\mathfrak{m}_{\boldsymbol x}=\langle T_{1}-x_{1},\dots,T_{1}-x_{n} \rangle$. > > An honorable mention would be [[Noether's normalization theorem]], on which each of these results crucially depends and which really takes 'the most work' of all to prove. ^theorem ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```