I was asked by my professor to prove the Fundamental Theorem of Algebra using the fact that for any function defined on the projective plane $deg(div(f))=0$. Now if we let $p(x,z)=\frac{\sum_{i=0}^na_ix^iz^{n-i}}{z^n}, a_i\in\mathbb{K}$ then every polynomial has a pole at the point of order $n$ at infinity $(x,z)=(1,0)$ and thus it must have $n$ roots.
The thing is this proof does not require $\mathbb{K}$ to be algebraically closed. So this would imply that for example in $\mathbb{P}^1(\mathbb{R})$ every polynomial of degree $n$ has $n$ roots. Could someone please explain to me where is the mistake?