The Fundamental Theorem of Algebra states that any nonconstant polynomial has a root in the complex plane. In fact, a degree n polynomial has n roots, if you count roots with multiplicity, but it suffices to show that there is at least one root. The Fundamental Theorem of Algebra was proved by Gauss in his doctoral dissertation.

To prove this using topology, we need the idea of the winding number of an oriented curve in the plane about the origin. (more…)