I want to solve the equation $x^2+z-1=0$ in the ring $\mathbb{C}[[z]]$ which is the set of formal power series over the variable $z$ over the complex numbers.
I also have to prove that there are no solutions in $\mathbb{C}[z]$.
I think that I should somehow write my equation with power series something like $a(z)x^2+b(z)x+c(z)$ for the first part but I'm not sure what to do after this and how to write it like that.
I also know that if $\alpha$ is a root in $\bar{f}(x)$ with multiplicity one, then $f(x)$ has a root $a(z)\in\mathbb{C}[[z]]$ with $a_0=\alpha$.
You are looking for a power series $f\in\mathbb C[[z]]$ such that $f(z)^2+z-1=0$. Write $f(z)=a_0+a_1z+\cdots$ and then identify the coefficients: $a_0^2=1$, $2a_0a_1=-1$, $2a_0a_2+a_1^2=0$, and so on. As you can see $a_0\ne0$ and then can find $a_1,a_2,\dots$ step by step.
If instead $f\in\mathbb C[z]$, then from $f(z)^2=1-z$ you get, by taking degrees, that $2\deg f=1$, a contradiction.