Degree and Ramification points of an holomorphic map between Riemann Surfaces The question is the following: we have an holomorphic map from $\Bbb P^1$ to  $\Bbb P^1$, defined by $f(z)=z^3-3z$. I need to find the degree and the ramification points and their orders, then verify the Riemann Hurwitz formula.
Attempt: I know there are $3$ zeroes of order $1$ each and a pole of order $3$ at infinity. Then the degree should be sum of the zeroes minus the poles. So should the degree be $0$ in this case? Also I get that there are no ramification points since the multiplicity of each zero is $1$, but I get a problem with the Riemann Hurwitz formula, So I know I am wrong. Help will be appreciated
 A: The degree is 3, and the ramifications points are two $z=1$ and $z=-1.$ So the total ramification points are 2. The genus of $\mathbb{P}^1=\hat{\mathbb{C}}=0$, therefore 
$$ 0=3(0-1)+1 +\frac{B}{2}=-3+1+2$$
Where B is the total branching number. 
A: The existing answer is incomplete at best.
The mapping $f\colon \mathbb{P}^1\rightarrow \mathbb{P}^1$ indeed has degree $3$, since $\sum_{p\in f^{-1}(\infty)}\operatorname{mult}_p(f)=\operatorname{mult}_\infty(f)=-\operatorname{ord}_\infty(f)=3$. Here the symbol $\operatorname{mult}_p(f)$ denotes the multiplicity of $f$ at $p$, while $\operatorname{ord}_\infty(f)$ denotes the order of $f$ at the pole $\infty$.
The mapping $f$ has three ramification points, namely $\pm 1$ and $\infty$.
As we have seen, the pole $\infty$ is a ramification point of $f$ of multiplicity $3$. To find the other ramification points, we differentiate $f(z)$ to obtain $f'(z)=3z-3$. The zeroes of $f'(z)$, namely $\pm 1$, are precisely the ramification points of $f$ in $\mathbb{C}$. The multiplicity of $f$ at a ramification point $p \in \mathbb{C}$ is the smallest positive natural number $n$ such that the $n$-th derivative of $f$ does not vanish at $p$, i.e. $f^{(n)}(p)\neq0$. Thus, the ramification points $\pm 1$ both have multiplicity $2$.
Let us verify the Riemann-Hurwitz formula. The Riemann sphere has topological genus $0$. Thus, the Riemann-Hurwitz formula in our example reads as
$$2\cdot0-2 \overset{!}{=} (2\cdot 0 -2)\cdot3+(2+1+1).$$ The equality clearly holds and we thus have verified the Riemann-Hurwitz formula in this example.

If you or any future reader has difficulties with calculating ramification points and their multiplicities of a non-constant holomorphic mapping $\mathbb{P}^1\rightarrow \mathbb{P}^1$ (which is always rational), I recommend reading Chapter II.$4$ of Rick Miranda's Algebraic Curves and Riemann Surfaces.
