Under revision for publication in the American Mathematical Monthly All comments are very much welcome

Niveau: Secondaire, Lycée

• dissertation

Under revision for publication in the American Mathematical Monthly. All comments are very much welcome! THE FUNDAMENTAL THEOREM OF ALGEBRA MADE EFFECTIVE: AN ELEMENTARY REAL-ALGEBRAIC PROOF VIA STURM CHAINS MICHAEL EISERMANN L'algebre est genereuse ; elle donne souvent plus qu'on lui demande. (d'Alembert) ABSTRACT. Sturm's famous theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any given real polynomial. In his residue calculus of complex functions, Cauchy (1831/37) extended this to an algebraic method to count and locate the complex roots of any given complex polynomial. We give a real-algebraic proof of Cauchy's theorem starting from the axioms of a real closed field, without appeal to analysis. This allows us to algebraically formalize Gauss' geometric argument (1799) and thus to derive a real-algebraic proof of the Fundamental Theorem of Algebra, stating that every complex polynomial of degree n has n complex roots. The proof is elementary inasmuch as it uses only the intermediate value theorem and arithmetic of real polynomials. It can thus be formulated in the first-order language of real closed fields. Moreover, the proof is constructive and immediately translates to an algebraic root-finding algorithm. The latter is sufficiently efficient for moderately sized polynomials, but in its present form it still lags behind Schonhage's nearly optimal numerical algorithm (1982).

• present approach


• extended sturm's


• algebraic proof


• real closed


• over


• any given rectangle


• all computation


• sturm's theorem