Proof that every rational algebraic equation has a root
A. C. Dixon · Proceedings of the Edinburgh Mathematical Society · 1907
The following arrangement of the proof of this theorem could, I think, be given at a comparatively early stage, even if the necessary case of De Moivre's theorem had to be proved as an introductory lemma. Let u, v be two rational integral algebraic functions of x, y with real coefficients, and let c be a simple closed contour in the plane. As the point (x, y) travels round c let those changes in the sign of u that take place when v is positive be marked and let (u, v; c) denote the excess in number among these of changes from + to − over changes from − to + *.