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)?