I'm studying Silverman's complex analysis but this book seems a lot slack. I have a question in page273:
Suppose $f(z)$ is a nonconstant analytic at $z_0$ with $f(z_0)$=0. Then, by the corollary to Theorem 8.13, there exists a neighborhood of $z_0$ that contains no other zeros of $f(z)$. Thus we may express $f(z)$ as $f(z)=(z-z_0)^k F(z)$ ($k$ a positive integer), where $F(z)$ is analytic at $z_0$ with no zeros in the neighborhood or on its boundary $C$.
I can't understand the last sentence(bold fonts). How can we know there is a factor $(z-z_0)^k$ and even more $F(z)$ is analytic?
I also have Ahlfors' book so you can give me references in Silverman or Ahlfors.