Ramification Locus and Ramification Index

Question

After searching for some examples on the site, I still don't quite get what how one calculates the ramification loci and indices for a morphism.

Here are a few examples that I think may help me understand. All of them are $\mathbb{P}^1\rightarrow \mathbb{P}^1$

1. $z+\frac{1}{z}$ This has ramification loci at 1 and -1, both with indices 2.

2. $\frac{z+2}{z-2}$ This is unramifed.

3. $z^3(z-1)^2$ This is an example from Silverman's AEC, and was also asked on this site several times such as here. It has ramification loci at 0 and 1 according to the answer.

I think I am looking for $a$ such that $f(z)-a=0$ has double (or more) roots, and go back to find $f^{-1}(a)$? Thus in general, I should be looking at the derivatives, to find where $f'(z)=0$? Then 1) and 2) would makes sense to me, but for 3), $f'(\frac{3}{5})=0$ as well, but it is not ramification loci.

Answer 1

The map in your third example is also ramified at $(3/5:1)$. The book says that the map is ramified at these points, not that these are all the points of ramification. From p. 3 of the errata:

This example is correct when it says that "$\phi$ is ramified at the points $[0, 1]$ and $[1, 1]$." These are the points in $\phi^{-1}([0, 1])$, so are the only relevant points for illustrating (II.2.6a). However, it is possibly a bit misleading to phrase it in this way, because $\phi$ has other ramification points. More precisely, it is also ramified at the points $[3/5, 1]$ and $[1, 0]$, which have ramification indices $2$ and $5$, respectively.