On surjectivity of smooth maps into Euclidean spaces and the fundamental theorem of algebra

Abstract: In this note we obtain the surjectivity of smooth maps into Euclidean spaces under mild conditions. As application we give a new proof of the Fundamental Theorem of Algebra. We also observe that any $C^1$-map from a compact manifold into Euclidean space with dimension $n\ge2$ has infinitely many critical points.