Question: Let $p(z)$ be a polynomial over $\mathbb{C}$. Is it true that $p:\mathbb{C} \to \mathbb{C}$ is a covering map ?
Partial answer: Let us look first at the points where $p'(z)\ne 0$. There the inverse function theorem tells us much more -- that $p$ is locally invertible, and the inverse is in fact holomorphic.
At the points where $p'(z) = 0$, however, $p$ will not be $1-1$ in any neighborhood, and certainly we cannot find an open neighborhood $U$ such that $p^{-1}(U)$ is a family of disjoint open sets, each homeomorphic to $U$.
Is the reasoning correct ? any comments are welcome.