Early proofs of Hilbert's Nullstellensatz

TL;DR

This paper reviews early proofs of Hilbert's Nullstellensatz, including Rabinowitsch's, Lasker's, and Hentzelt's approaches, highlighting their historical development in commutative algebra.

Contribution

It provides a comprehensive analysis of the initial proofs of Nullstellensatz and their significance in the evolution of algebraic theory.

Findings

Rabinowitsch's proof derives Nullstellensatz from the weak version.

Lasker's proof uses primary decomposition.

Historical context of proofs up to van der Waerden's work.

Abstract

By Rabinowitsch' trick Hilbert's Nullstellensatz follows from the weak Nullstellensatz (Rabinowitsch 1929). The weak version can be shown with elimination theory. Hilbert's original proof is also based on successive elimination. Lasker obtained a new proof using primary decomposition. We describe these early proofs and place them in the development of commutative algebra up to the appearance of van der Waerden's Moderne Algebra. We also explain Hentzelt's Nullstellensatz.