On the BCSS Proof of the Fundamental Theorem of Algebra

Chapter 10 of the 1998 Springer-Verlag book { Complexity and Real Computation}, by Blum, Cucker, Shub, and Smale, contains a particularly elegant proof of the Fundamental Theorem of Algebra: The central idea of the proof naturally leads to a homotopy continuation algorithm for finding the roots of univariate polynomials, and extends naturally to a proof B\'{e}zout's Theorem (on the number of roots of systems of n equations in n unknowns). We present a more detailed version of the BCSS Proof which is hopefully useful for those new to algebraic geometry. Any errors here are solely the responsibility of the current author.