I'm trying to understand the proof of the Fundamental Theorem of Algebra in Theorem 3.7 here.
I can't get my head around this sentence though
If $p(z)$ has no roots at all, the map $p|_{S1(R)}$ factors through the complex plane $\mathbb{C}$ and is therefore nullhomotopic (as $\mathbb{C}$ is contractible).
I know that $\mathbb{C}$ is convex so if $p$ is a map into $\mathbb{C}$ then it is homotopic to any other map into $\mathbb{C}$. Does this help? Also surely any map into $\mathbb{C}\setminus\{0\}$ is a map into $\mathbb{C}$. Doesn't this make the whole thing vacuous?!
Many thanks in advance.