A bottom-up approach to Hilbert's Basis Theorem

TL;DR

This paper provides an accessible explanation of commutative algebra concepts, introduces polynomial rings, and presents a proof of Hilbert's Basis Theorem, connecting algebraic structures to geometric properties.

Contribution

It offers an expository account of Hilbert's Basis Theorem with detailed proofs and insights into algebraic geometry foundations.

Findings

Proof of Hilbert's Basis Theorem included

Clarification of polynomial rings and their geometric significance

Enhanced understanding of algebraic structures and ideals

Abstract

In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we introduce the polynomial ring and the fundamental relationship between polynomials and sets of points. We prove some results in algebraic geometry, notably Hilbert's Basis Theorem.