Nullstellen and Subdirect Representation

摘要

David Hilbert's solvability criterion for polynomial systems in n variables from the 1890s was linked by Emmy Noether in the 1920s to the decomposition of ideals in commutative rings, which in turn led Garret Birkhoff in the 1940s to his subdirect representation theorem for general algebras. The Hilbert-Noether-Birkhoff linkage was brought to light in the late 1990s in talks by Bill Lawvere. The aim of this article is to analyze this linkage in the most elementary terms and then, based on our work of the 1980s, to present a general categorical framework for Birkhoff's theorem.