Edit: I know that Euler had a method, which was incomplete, then Gauss proved the FTA. If somebody could show me what Euler did, it would be great. I think that's what I need.
The Aim is to prove the FTA, but using a incomplete proof. I'm given a polynomial $P(z)$ with degree $n$.
I'm told to consider $P(re^{i\theta})$ where $0\leq \theta\leq 2\pi$, fix $r$.
I need to argue that there exist $r,\theta$ such that $P(re^{i\theta})=0$, but this proof is non-complete, why?
My thoughts:
I know there is a circle for all $r$. Now this circle takes complex values. And I need to map this circle in complex numbers, apply $P$ to it, and see what values the polynomial takes on. Then I can apply intermediate value theorem to find roots.
I also thought about the existence of such number. Every polynomial has a root (I'm not saying there are exactly $n$ roots) $\Rightarrow$ I can always parametrize my root to find such $r,\theta$
There is a nice text of W. Dunham about Euler and the Fundamental Theorem of Algebra, which could be what you need.