Real Polynomial Rings and Domain Invariance

Recent proofs of classical theorems in polynomial algebra and functional analysis are discussed, which use tools from the topology of real manifolds. Simpler proofs were discovered in the new century, of the Hilbert Nullstellensatz, and the Gelfand-Mazur Theorem. We give a related proof that an irre...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Author Sjogren, Jon A
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 01.02.2015
Subjects
Fixed points (mathematics)
Functional analysis
Invariance
Manifolds (mathematics)
Mapping
Polynomials
Rings (mathematics)
Theorems
Topology
Online AccessGet full text

Cover

More Information
Summary:Recent proofs of classical theorems in polynomial algebra and functional analysis are discussed, which use tools from the topology of real manifolds. Simpler proofs were discovered in the new century, of the Hilbert Nullstellensatz, and the Gelfand-Mazur Theorem. We give a related proof that an irreducible real polynomial has degree 2 or less, Gauss's form of the Fundamental Theorem of Algebra. It has been debated whether an elementary proof for FTA can be found, using the Brouwer Fixed-Point Theorem as its "analytical" component. In the present case the analytic or topological tool employed is Brouwer's Theorem on Invariance of Domain, which derives from his Fixed-Point Theorem. A corollary of Domain Invariance is that an injective mapping of one compact manifold to another (connected) one of the same dimension, is in fact surjective and a homeomorphism. The desired result (FTA) comes from the fact that a real sphere and its (quotient) projective space of the same dimension are homeomorphic only when this dimension equals 1. This proof joins a class of proofs that depend on Euclidean fixed-point theory, and also the class of proofs that involve no field extensions or methods of complex analysis.
ISSN:2331-8422