I am trying to make a proof of Fundamental Theorem of Algebra which is seem to be understandable by secondary school student. Although I use some topological argument, but I think I could cheat them except the point I have mentioned.

What I use about complex is open mapping theorem and is a holomorphic function.

Let be a polynomial which is non-constant. It can extend to a continuous function, also denoted as , by such that maps to . It is well-defined and continuous by using limit. By open mapping theorem (can be replaced by elemantary argument?)(1), is open ( for the neighborhood of , , observe that (1) is open and (2) contain the set for some small $r$ (or by continuity of ), is open) . Also, is closed. By connectness, , hence there exists such that .

(1) can be reduced to show that is open.

But still, I use open mapping theorem. Can I use more elementary way to show (1)?

It seems it is not need to that 1/f is holomorphic.

We need it. The crucial point is that is open, even at the point of infinity. How can we show this?

The continuity of (or 1/f) is not enough, what we need is the open mapping theorem (applied to 1/f), which is a nontrivial result.

By the way, this is KKK. I am now in Shanghai.