Hilbert Nullstellensatz And Dimension of a Polynomial Ring

Majumder, Soumak (2025) Hilbert Nullstellensatz And Dimension of a Polynomial Ring. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Soumak Majumder (23MP017)) 23MP017_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (8MB)

Abstract

The main theorems of this thesis are Hilbert Nullstellensatz , which has two version one is Weak Hilbert Nullstellensatz and another is Strong Hilbert Nullstellensatz , other is Dimension of a polynomial Algebra . In weak Hilbert Nullstellensatz we see how the maximal ideals in a polynomial ring over an algebraically closed field looks like . Then for the strong version we define Variety of a subset and Zero ideal of a set and study some properties regarding the Variety and Zero ideal , and then by the help of Strong Hilbert Nullstellensatz we shown there is a one-one correspondence between Variety and the zero ideals of a set . And for the other part we define Krull dimension and height of an ideal and some example . Finally we present the main theorem about the dimension of a polynomial algebra over a noetherian ring .

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