Denote by $K=\mathbb{C}((z))$ the fraction field of $\mathbb{C}[[z]]$.
Define an embedding of $K$ onto itself taking $a(z)$ to $a(z^n)$ $\forall n$. The target is $\mathbb{C}((z^{1/n}))$. Define the formal field of Puiseux series as the union of $\mathbb{C}((z^{1/n}))$ and denote it by $\mathbb{C}((z^*))$.
Using the Hensel Lemma prove that any polynomial $P(x)\in K[x]$ of degree greater than one is a product of two polynomials with coefficients in $\mathbb{C}((z^*))$ of positive degree.
Prove that the field of formal Puiseux series $\mathbb{C}((z^*))$ is algebraically closed.
For 1. It tells me to use Hensel's Lemma but I am not sure how to solve it.
For 2. I guess that I should use 1., but I can't quite get it right in my head. I have also heard that Newton's Method can solve this problem, but I don't understand any of what I have read about it.