[Paper Review] Real Polynomial Rings and Domain Invariance

Jon A. Sjogren|arXiv (Cornell University)|Feb 1, 2015

Homotopy and Cohomology in Algebraic Topology8 references3 citations

TL;DR

This paper proves the Fundamental Theorem of Algebra using topological methods centered on Brouwer's Invariance of Domain theorem. By constructing a continuous, injective map from real projective space ℝPⁿ⁻¹ to the sphere 𝕊ⁿ⁻¹ via squaring in a real algebra derived from an irreducible polynomial, the authors show that such a map can only be a homeomorphism if n ≤ 2, thereby proving that irreducible real polynomials must have degree at most 2.

ABSTRACT

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.

Motivation & Objective

• To provide a topological proof of the Fundamental Theorem of Algebra using tools from algebraic topology.
• To demonstrate that irreducible real polynomials must have degree ≤ 2 by analyzing the topological structure of associated algebras.
• To establish a connection between the algebraic structure of real polynomial quotient rings and topological invariants like homotopy and connectedness.
• To show that the existence of an irreducible real polynomial of degree n ≥ 3 leads to a topological contradiction via the Invariance of Domain theorem.

Proposed method

• Construct the quotient ring 𝒪 = ℝ[x]/(p(x)) for an irreducible real polynomial p(x) of degree n, which forms a real algebra without zero divisors.
• Define a squaring map f(a) = a∘a on 𝒪, then normalize it to obtain a continuous map ψ: 𝕊ⁿ⁻¹ → 𝕊ⁿ⁻¹.
• Lift ψ to a map ̂ψ: ℝPⁿ⁻¹ → 𝕊ⁿ⁻¹ by exploiting the symmetry ψ(y) = ψ(−y), which respects the antipodal equivalence.
• Prove that ̂ψ is continuous, injective, and between compact Hausdorff spaces, hence a homeomorphism onto its image.
• Apply Brouwer’s Invariance of Domain theorem to show that the image of ̂ψ must be open and closed in 𝕊ⁿ⁻¹, hence surjective when 𝕊ⁿ⁻¹ is connected (for n > 1).
• Derive a contradiction by constructing a non-contractible loop in ℝPⁿ⁻¹ (via the universal cover) that would have to lift to a homotopy with fixed endpoints, contradicting the contractibility of loops in 𝕊ⁿ⁻¹ for n > 2.

Experimental results

Research questions

• RQ1 Can the Fundamental Theorem of Algebra be proven using topological methods rather than purely algebraic or analytic ones?
• RQ2 What topological constraints arise from assuming the existence of an irreducible real polynomial of degree n ≥ 3?
• RQ3 How does the squaring map on a real algebra derived from a polynomial induce a continuous map between spheres and projective spaces?
• RQ4 In what way does the Invariance of Domain theorem obstruct the existence of higher-degree irreducible real polynomials?
• RQ5 Why is the real projective space ℝPⁿ⁻¹ not homeomorphic to the sphere 𝕊ⁿ⁻¹ for n > 2, and how does this relate to polynomial irreducibility?

Key findings

• An irreducible real polynomial must have degree at most 2, as any higher degree leads to a topological contradiction.
• The map ̂ψ: ℝPⁿ⁻¹ → 𝕊ⁿ⁻¹ induced by squaring in the algebra is a homeomorphism if and only if n ≤ 2.
• The existence of a non-contractible loop in ℝPⁿ⁻¹ that cannot be homotoped to a constant loop, despite the sphere being simply connected, leads to a contradiction for n > 2.
• The Invariance of Domain theorem is essential in proving that the image of ̂ψ is both open and closed, hence surjective, when the target space is connected.
• The proof relies on the unique path-lifting property of the universal cover and the fact that antipodal points in 𝕊ⁿ⁻¹ project to the same point in ℝPⁿ⁻¹.
• The contradiction arises because a loop in ℝPⁿ⁻¹ that connects the poles cannot be contracted without violating the endpoint-fixing property of lifts under the universal cover.