Math for freedom. An original proof of the fundamental theorem of algebra within the ambit of real numbers

At the beginning of my work there’s a description of a particular philosophy of mathematics, the mathematics for freedom, which is used to ultimately justify thoughts and positions about the didactics of mathematics. The main remark is that didactics of mathematics won’t be incisive by remaining on general considerations and thus avoiding the specific features of each topic to be learned. The origin of the didactic hints relative to specific topics can only derive from a profound mathematical study of these topics. The last observation is naturally followed by an explanation and a classification of what a profound mathematical study of a topic is. For many elementary topics in mathematics there is not enough mathematical research already performed at high levels of profundity, and this causes a serious didactic difficulty. For this reason, didactical concerns can motivate mathematical research. This is the case of the fundamental theorem of algebra in the ambit of real numbers. The existing proofs of this result make use of the complex numbers. The use of complex numbers causes that the fundamental ideas on which the proofs rely become very difficult to be identified. This motivates the development of an original proof of this result that avoids the use of complex numbers. As expected the proof I developed enlightens on the fundamental ideas on which the result rests. The zero-level curves that correspond to the remainder of the division between a generic even-degree polynomial and a quadratic polynomial present an interweaved pattern in a region far away enough from the origin, and this implies the existence of an intersection of these zero-level curves and therefore also the existence of a quadratic polynomial dividing the given even-degree polynomial. The proof makes extensive use of the recursive properties of the algebraic expression of the remainder and of continuity.