= Winding-number proof of the fundamental theorem of algebra
For a polynomial $P(z)=a_nz^n+\cdots+a_0$ of positive degree, on a sufficiently large circle its leading term dominates the remaining terms. The <dominated perturbation preserves winding number>[dominated-perturbation lemma] therefore gives winding number $n$ to the loop $P(Re^{2\pi it})$. If $P$ had no zero, radial contraction of the input circle would map under $P$ to a homotopy with a constant loop in $\mathbb C\setminus\{0\}$, which has winding number zero. This contradiction proves the <Fundamental theorem of algebra>.
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