Degree of a power map of the Riemann sphere ($\mathrm{deg}(\zeta \mapsto {\zeta }^k)=k$)

For $k\ge 1$, the power map has $k$ distinct preimages of every nonzero finite regular value, all with positive local degree of a map between oriented manifolds. Integrating the pullback of a differential form of $\Omega =id\zeta \wedge d\bar{\zeta }/(1+|\zeta {|}^2{)}^2$ gives the same answer, since $\int \Omega =2\pi$ and $\int f^{\ast }\Omega =2\pi k$. For $k=0$ use the constant extension $1$ at infinity; its degree of a map between oriented manifolds is zero.