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A one-line undergraduate proof of Zariski's lemma and Hilbert's nullstellensatz
| Content Provider | Semantic Scholar |
|---|---|
| Author | Azarang, Alborz |
| Copyright Year | 2015 |
| Abstract | The first part of this theorem is Zariski’s Lemma and the second part is usually called Hilbert’s Nullstellensatz (Weak Form), which says every maximal ideal M of the polynomial ring K[x1, x2, . . . , xn], where K is an algebraically closed field is of the form M = (x1 − a1, x2 − a2, . . . , xn − an), ai ∈ K for all i. Usually the proof of Zariski’s Lemma depends on two technical lemmas due to Artin-Tate and Zariski, see [7, Proposition 3.2, and the comment following it]. For an elegant proof of Hilbert’s Nullstellensatz, based on G-ideal theory, see [5]. We should also remind the reader that in some elementary text books on Algebraic Geometry, Noether normalization theorem is applied for the proof of Zariski’s Lemma, see for example [4, Theorem 1.15] and [8]. Two different proofs of this lemma are also given in [11, P. 166]. Before presenting our proof, we should emphasize here that, it seems, the next proof is the simplest and the shortest possible proof (among the existing proofs) of the results in the title, for now. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/1506.08376v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
