Connectivity of Complex Analysis Polynomial Sets | Degree n+1

[Mystic998]

Homework Statement

Let $p(z)$ be a polynomial of degree $n \geq 1$. Show that [itex]\left\{z \in \mathbb{C} : \left|p(z)\right| > 1 \right\}[/tex] is connected with connectivity at most $n+1$.

Homework Equations

A region (connected, open set) considered as a set in the complex plane has finite connectivity n if its complement has n connected components in the extended complex plane.

The Attempt at a Solution

I'm not sure where to start, frankly. Showing the set is connected seems really tricky, though I'm admittedly probably overlooking something really obvious. As for connectivity, I think it has to do with the fact that the complement of the set is $\left\{z \in \mathbb{C} : \left|p(z)\right| \leq 1\right\} \cup \left\{\infty\right\}$ in the extended complex plane. So I think that because the polynomial has at most n roots, any preimage of the first set can have at most n disjoint connected sets mapped to it, then the point at infinity gives you one more connected component. But I'm not sure how to say that rigorously.

 

[Mystic998]

Bump before bed