Degree of the map $S^1\to S^1:z\to z^n$

Question

I want to compute the (topological/differential) degree of the map $f:S^1\to S^1:z\mapsto z^n$.

I've have shown that the degree of the map $g:\mathbb{C}^*\to \mathbb{C}^*:z\mapsto z^n$ is $n$. Is there a way to deduce $\text{deg}(f)$ from $\text{deg}(g)$? What are other ways to compute $\text{deg}(f)$?

Answer 1

This depends strongly on your definition of the degree, but here is an attempt:

The preimage of a value on $S^1$ under $f$ is $n$ points. All these points are regular, and they all preserve the orientations of the $S^1$'s, hence contribute $+1$ to the degree. Summing these up one sees that $f$ is a degree $n$ map.