Regular points, Fundamental Theorem of algebra

Let $f: M \to N$ be a smooth map between two manifolds. We call $x \in M$ a regular point provided the derivative $Df_x$ is non singular, so that there exists a neighbourhood of $x$ such that it is diffeomorphically mapped onto its image.(See Inverse function theorem) A point $y \in N$ is called a regular value if $f^{-1}(y)$ contains only regular points. Otherwise we call them crtitical points and values respectively.

Now if $M$ were a compact manifold, then $f^{-1}(y)$ for every regular value $y$ is a finite set. This is because $f^{-1}(y)$ , being a closed set of compact set $M$ , is compact and is discrete because neighbourhoods of the regular points in $f^{-1}(y)$ are mapped one-one to $N$ .

So for every regular value $y$ we can define $n(f^{-1}(y))$ to be the number of regular points corresponding to $y$ . This function on the set of regular values of $N$ is a locally constant function i.e., there exists a neighbourhood of $y$ such that for every $y_1$ in that neighbourhood $n(f^{-1}(y_1))=n(f^{-1}(y))$ . Choose the neighbourhoods of points in $f^{-1}(y)$ which diffeomorphically mapped onto their images. Properly restricting the images which are neighbourhoods of $y$ will give you the required neighbourhood of $y$ such that $n$ is locally constant.

Now we shall use the above results to give proof the fundamental theorem of algebra. To apply the above result we start by compactifying the complex plane. Every polynomial function which is a smooth function from the complex plane to itself can be seen as a function on the Riemann sphere to itself by using the stereographic projections for transition between the complex plane and the sphere. If P is the polynomial, the function on the sphere is given by $f(x)= h^{-1}Ph(x)$ for $x \ne \mbox{north pole}$ , $f(\mbox{north pole})= \mbox{north pole}$ where $h: \mathbb{C}_\infty \to \mathbb{C}$ is a smooth transition function from the Riemann sphere to Complex plane.

This function is infact smooth on the whole sphere. ( To see that it is smooth at poles use stereographic projections from south pole also and argue.)

The function on the Riemann sphere now has critical values only at a finite number points. This is because the derivative of $P(z)$ $P'(z)$ vanishes atmost at $n-1$ points where n is the degree of $P(z)$ . Thus the set of regular values of $f$ is the sphere minus finitely many points. So it is connected, as a consequence of which the function $n(f^{-1}(y))$ which is locally constant function is actually a constant function now. It cant be zero everywhere. So $f$ has to be onto which implies that polynomial takes every value. $QED$