(1.20) Define the loop $\gamma_R(t)=p(Re^{i2\pi t})$. If $p$ has no complex root then $\gamma_R$ is a loop in $\mathbf{C}\setminus\{0\}$. We will prove that the degree $n$ of $p$ is equal to zero.

(2.40) As $R$ varies, this gives a free homotopy. The loop $\gamma_0$ is the constant loop at $p(0)$.

(3.20) Since $\pi_1(\mathbf{C}\setminus\{0\},x$ is abelian, free and based homotopy agree, so we get $[\gamma_0]=[\gamma_R]\in\pi_1(\mathbf{C}\setminus\{0\},x)=\mathbf{Z}$. Since $\gamma_0$ is constant, $[\gamma_0]=0$.

(4.50) We will show that, for large $R$, $[\gamma_R]=n\in\mathbf{Z}=\pi_1(\mathbf{C}\setminus\{0\})$. This will imply

$$n=0.$$ The loop $\delta_n(t):=R^ne^{i2\pi nt}$ satisfies $[\delta_n]=n\in\mathbf{Z}$, so we need to show $\gamma_R\simeq\delta_n$ for large $R$.

(6.24) To achieve this, write $p(z)=z^n+q(z)$ ($q(z)=a_{n-1}z^{n-1}+\cdots+a_0$) and try the homotopy $H(s,t)=R^ne^{i2\pi nt}+sq(Re^{i2\pi t})$, which connects $\delta_n$ at $s=0$ to $\gamma_R$ at $s=1$. We need this to be a homotopy in $\mathbf{C}\setminus\{0\}$, so we need to show that $|H(s,t)|>0$ for all $s,t$ when $R\gg 0$.

(9.00) We estimate:

$$\begin{align*} |H(s,t)|&\geq ||R^ne^{i2\pi nt}|-s|q(Re^{i2\pi t})||\\ &\geq |R^n-R^{n-1}\max_{k=0}^{n-1}|a_k||, \end{align*}$$ (10.30) using $$\begin{align*} |q(Re^{i2\pi t})|&=|a_{n-1}R^{n-1}e^{2\pi i(n-1)t}+\cdots+a_0|\\ &\leq |a_{n-1}|R^{n-1}+\cdots+|a_0|\\ &\leq nR^{n-1}\max_{k=0}^{n-1}(|a_k|). \end{align*}$$ (12.08) so $$\begin{align*} |H(s,t)|&\geq R^{n-1}(R-\max_{k=0}^{n-1}|a_k|), \end{align*}$$ which is strictly positive whenever $R$ is strictly greater than $\max_{k=0}^{n-1}(|a_k|)$.

(13.00) This implies that $H(s,t)$ is a homotopy in $\mathbf{C}\setminus\{0\}$ between $\gamma_0$ (with $[\gamma_0]=0$) and $\gamma_R$ (with $[\gamma_R]=n$), so $n=0$ and $p$ is constant, which completes the proof.