In the proof of Rouche's theorem I saw in a book, there are two points I failed to understand, or failed to prove myself. (if you aren't familiar with the theorem, please try to look at the two statements here below and explain them regardless of the theorem):
"Consider the complex plane. It's not difficult to show that the index of a curve C with respect to a vector field v is equal to the sum of indices of singular points, i.e., those at which v(z) = 0."
"If f(z) is a polynomial, and let v(z) = f(z), where v is a vector field, then the index of the singular point z0 is equal to the multiplicity of the root z0 of f."
I am having trouble understanding this, can anyone please help?