A topological proof for this statement relies on the contradiction that two curves with different winding numbers would have a homotopy if this statement were not true. Since Liouville's theorem implies that bounded, analytic
A topological proof for this statement relies on the contradiction that two curves with different winding numbers would have a homotopy if this statement were not true. Since Liouville's theorem implies that bounded, analytic functions are constant, it can be used to prove this statement. An early proof for it came from Gauss' doctoral dissertation, and it can be equivalently stated as "the field of complex numbers is closed." It states that for a polynomial of degree m with complex coefficients, the polynomial has at most m roots and at least one complex root. For 10 points, identify this statement which implies that a polynomial can be written as a product of linear factors. 
A topological proof for this statement relies on the contradiction that two curves with different winding numbers would have a homotopy if this statement were not true.
{fundamental} theorem of {algebra} [prompt on {fta} or other {such lazy attempts}]