Strictly Real Fundamental Theorem of Algebra
Soham Basu
TL;DR
This paper provides an accessible proof of the Fundamental Theorem of Algebra for real polynomials, demonstrating the existence of quadratic factors using only basic real analysis principles.
Contribution
It offers a new, elementary proof of the theorem that avoids complex analysis, making it accessible to those with only fundamental real analysis knowledge.
Findings
Existence of real quadratic factors for any real polynomial.
Proof relies solely on basic real analysis concepts.
Approachable proof suitable for educational purposes.
Abstract
Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound property for real numbers, continuity and growth property of polynomials, and unfamiliar with complex numbers, field extension or advanced topology.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsHistory and Theory of Mathematics · Polynomial and algebraic computation · Mathematics and Applications
††affiliationtext: Corresponding author: [email protected]
Strictly Real Fundamental Theorem of Algebra
Soham Basu
Abstract
Without resorting to complex numbers [basu] or advanced topological arguments [Pukhlikov, Pushkar], we show that any real polynomial of degree has a real quadratic factor, which is equivalent to the seminal version of the Fundamental Theorem of Algebra (FTA) [Gauss]. Thus it is established that basic real analysis suffices as the minimal platform to prove FTA.
Keywords: Fundamental Theorem of Algebra, Polynomial Interlacing.