# 1.06 Fundamental theorem of algebra: reprise

## Video

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

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## Notes

### 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:

*(10.30)* using

*(12.08)* so

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

- 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.

## Navigation

- Previous video:
**1.05 Basepoint dependence**. - Next video:
**1.07 Induced maps**. - Index of all lectures.