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Sturm's theorem (1829/35) provides an elegant algorithm to count and locatethe real roots of any real polynomial. In his residue calculus (1831/37) Cauchyextended Sturm's method to count and locate the complex roots of any complexpolynomial. For holomorphic functions Cauchy's index is based on contourintegration, but in the special case of polynomials it can effectively becalculated via Sturm chains using euclidean division as in the real case. Inthis way we provide an algebraic proof of Cauchy's theorem for polynomials overany real closed field. As our main tool, we formalize Gauss' geometric notionof winding number (1799) in the real-algebraic setting, from which we derive areal-algebraic proof of the Fundamental Theorem of Algebra. The proof iselementary inasmuch as it uses only the intermediate value theorem andarithmetic of real polynomials. It can thus be formulated in the first-orderlanguage of real closed fields. Moreover, the proof is constructive andimmediately translates to an algebraic root-finding algorithm.

The Fundamental Theorem of Algebra made effective: an elementary real-algebraic proof via Sturm chains