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...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
01.02.2015
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Subjects | |
Online Access | Get full text |
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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. |
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DOI: | 10.48550/arxiv.1502.01037 |