An Introduction to Hilbert’s Nullstellensatz: an Insight for the Algebraically Minded
This paper contains Hilbert's Nullstellensatz, one of the fundamental theorems of algebraic geometry. The main approach uses the algebraic proof and all of the necessary background to understand and appreciate the Nullstellensatz. The paper also contains the Combinatorial Nullstellensatz, the proof and some applications. This thesis is aimed at any science student who wants to understand Hilbert's Nullstellensatz or gain intuitive insight. The paper starts with an explanation of the approach: the theorem is first presented in its algebraic form and the proof follows from the given background. After that, there are some more geometric definitions relating to common zero sets or varieties, and then Hilbert's Nullstellensatz is given in a slightly different manner. At the end of the paper some combinatorial definitions and theorems are introduced so that the Combinatorial Nullstellensatz can be proven. From this follows the Cauchy-Davenport theorem. At the very end of the introduction is presented a case of the theorem for the readers who do not yet know what Hilbert's Nullstellensatz is about. The second section of the paper contains all of the algebraic background material. It starts from the most basic definitions of groups and homomorphisms and then continues onto rings, ideals, radicals and quotient rings and relations between them. Some additional theorems are marked with a star to denote that the statement is not necessary for the algebraic proof of Hilbert's Nullstellensatz, but it might be necessary for the geometrical or combinatorial proofs. Since these statements are fully algebraic and mostly contain definitions that have been introduced early in the paper, they are placed in the algebraic section near similar theorems and definitions. The second section also contains fields, algebras, polynomials and transcendental elements. In Sections 3 and 4 we follow the algebraic proof of Daniel Allcock and expand on some steps. Section 3 contains Zariski's Lemma and a form of the Weak Nullstellensatz, along with their proofs. These statements allow us to skip Noetherian and Jacobson rings by finding the necessary conclusions from polynomial rings over fields. This section also includes an existence conclusion that also can be found in other literature under the name of Weak Nullstellensatz. Afterwards, Section 4 follows Allcock's proof of Hilbert's Nullstellensatz, working from the Weak Nullstellensatz and applying the Rabinowitsch trick. Section 5 explains how the Nullstellensatz brings together algebra and geometry. It is split up into three parts: first some preliminary definitions and theorems are defined. One of the fundamental definitions is a variety, which is simply the common zero set of some polynomials. After the new definitions we again meet Hilbert's Nullstellensatz and show that it is equivalent to the previous form. Using the newfound equivalence we show that varieties constitute a topology and consider the dimension of such spaces. Lastly, we show the equivalence of algebra homomorphisms and the corresponding variety morphisms. Section 6 slightly diverges and considers the Combinatorial Nullstellensatz introduced by Noga Alon. This theorem resembles Hilbert's Nullstellensatz, yet is different when looked at carefully. We consider the original proof by Alon, and additionally show how the Combinatorial Nullstellensatz follows from Hilbert's Nullstellensatz, using the paper by Goel, Patil and Verma. We conclude the paper by proving the Cauchy-Davenport theorem, using the Combinatorial Nullstellensatz. This paper does not delve deep in any particular topic. Quite the contrary, it connects many different theorems and shows equivalences between them while referencing additional reading material.
Nullstellensatz, algebraic geometry, algebra, variety, combinatorics, homomorphism
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