The Fundamental Theorem of Algebra: An Elementary and Direct Proof

@article{Oliveira2011TheFT,
  title={The Fundamental Theorem of Algebra: An Elementary and Direct Proof},
  author={Oswaldo Rio Branco de Oliveira},
  journal={The Mathematical Intelligencer},
  year={2011},
  volume={33},
  pages={1-2}
}
  • O. Oliveira
  • Published 4 March 2011
  • Mathematics
  • The Mathematical Intelligencer
We present a simple, differentiation-free, integrationfree, trigonometryfree, direct and elementary proof of the Fundamental Theorem of Algebra. “The final publication (in The Mathematical Intelligencer, 33, No. 2 (2011), 1-2) is available at www.springerlink.com: http://www.springerlink.com/content/l1847265q2311325/” 
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References

SHOWING 1-10 OF 22 REFERENCES
Mathematics and its history
Preface.- The Theorem of Pythagoras.- Greek Geometry.- Greek Number Theory.- Infinity in Greek Mathematics.- Polynomial Equations.- Analytic Geometry.- Projective Geometry.- Calculus.- Infinite
On the Fundamental Theorem of Algebra
One of the simplest proofs that every nontrivial polynomial P has a zero goes as follows. Observe that |P(z)| → ∞ as |z| → ∞, so we may find an R > 0 with |P(z)| > |P(0)| for all |z| ≥ R. Since any
Elements of Abstract Analysis
1. Sets.- 1.1 Set Theory.- 1.2 Relations and Functions.- 1.3 Ordered Sets.- 1.4 Ordinals.- 1.5 The Axiom of Choice.- 2. Counting.- 2.1 Counting Numbers.- 2.2 Cardinality.- 2.3 Enumeration.- 2.4
Principles of mathematical analysis
Chapter 1: The Real and Complex Number Systems Introduction Ordered Sets Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces Appendix Exercises Chapter 2: Basic
Fubinito (Immediately) Implies FTA
Example 2. Given a semiregular octagon AxA^^A^AsA^AqA^, we consider its en veloping quadrilaterals BXB2B3B4 and CXC2C3C4 determined by the lines containing the sides AXA2, A3A4, A5A6, A7AS, and A2A3,
Philosophie mathématique. Réflexions sur la nouvelle théorie des imaginaires, suivies d'une application à la démonstration d'un théorème d'analise
© Annales de Mathématiques pures et appliquées, 1814-1815, tous droits réservés. L’accès aux archives de la revue « Annales de Mathématiques pures et appliquées » implique l’accord avec les
Chapters 3 and 4
  • Icrp
  • Environmental Science
  • 2007
Mathematics and its History, Springer-Verlag
  • New York,
  • 1989
...
...