1.06 Fundamental theorem of algebra: reprise


Below the video you will find accompanying notes and some pre-class questions.


Fundamental theorem of algebra

(0.00) We've now developed enough technology to come back and prove the fundamental theorem of algebra rigorously. The only fact we'll need to assume from later in the module is that \[\pi_1(\mathbf{C}\setminus\{0\},x)\cong\mathbf{Z},\] and that the loop \(\delta_n(t):=e^{i2\pi nt}\) satisfies \([\delta_n]=n\in\mathbf{Z}\).

(0.33) A nonconstant complex polynomial \(p(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_0\) has a complex root, i.e. \(p(z)=0\) for some \(z\in\mathbf{C}\).

(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 R^{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.

Pre-class questions

  1. If you're anything like me, that sequence of inequalities sounded something like ``blah blah blah blah blah blah blah''. Go back and look at them, and see if you can justify each step. If there's a step you can't justify, make a note of it and we can check it in class.


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