Show that the map $p:\mathbb{C}\rightarrow \mathbb{C}$ where $p(z)=z^m+a_1z^{m-1}+...+a_{m-1}z+a_m$ is a submersion except at finitely many points.
My Attempt:
I calculated $df_{z_0}(w)$ and finally got that $\displaystyle df_{z_0}(w)=\lim_{t\rightarrow 0} \frac{p(z_0+tw)-p(z_0)}{t}=a_{m-1}w+kw$ where $k$ is a constant that depends on $z_0$. So, I think $df_{z_0}(w)$ is surjective even when $z_0=0$. But why does it ask to show it is surjective only except finitely many points? Am I doing something wrong with my calculation? Please help.
EDIT: I think I got it. There might me $z_0's$ where $a_{m-1}+k=0$, and in that case not surjective. Is it correct?
The derivative of $p$ is a complex polynomial. Hence, it has a finite amount of points where it is zero (assuming we didn't begin with a constant polynomial, otherwise the proposition you want to prove isn't true). Since the derivative at any given point is a complex number, being non-zero makes it surjective as a linear map. So except on the roots of the derivative, the map is a submersion.