# 1.01 Winding numbers and the fundamental theorem of algebra

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

- Next video:
**1.02 Paths, loops and homotopies**. - Index of all lectures.

# Notes

*(0.00)* I want to start this module by giving a rough sketch of how
to prove the fundamental theorem of algebra using the idea of winding
numbers. You may have seen something similar in a first course on
complex analysis, where the winding number was defined using a contour
integral. The aim of the current proof is to remove the need for
complex analysis in this definition: the winding number is something
purely topological.

## The fundamental theorem of algebra

*(1.45)*Let \(p\colon\mathbf{C}\to\mathbf{C}\), \(p(z)=z^n+a_{n-1}z^{n-1}+\cdots+a_0\), be a complex polynomial. Assume that \(p\) has no complex root, in other words that there is no point \(x\in\mathbf{C}\) for which \(p(x)=0\). We will show that \(n=0\), in other words that \(p\) has only a constant term.

*(2.24)* Consider the circle of radius \(R\) in the complex
plane. The points in this circle are precisely those of the form
\(Re^{i\theta}\). Let \(\gamma_R\) be the image of this circle under
the map \(p\colon\mathbf{C}\to\mathbf{C}\). We can think of
\(\gamma_R\) as a loop in \(\mathbf{C}\):
\[\gamma_R(\theta)=p(Re^{i\theta}).\] Crucially, because \(p(z)\neq
0\) for all \(z\in\mathbf{C}\), the loop \(\gamma_R\) is a loop in
\(\mathbf{C}\setminus\{0\}\).

**Figure 1.**

*In this figure, we see the circle of radius \(R\) in the domain of \(p\) and its image \(\gamma_R\) in the image of \(p\) (\(\mathbf{C}\setminus\{0\}\)). In this example, \(p(z)=z^3-z/2\) and \(R=2\) and we see that the winding number of \(\gamma_R\) is 3.*

*(3.54)* When \(R=0\), \(\gamma_0(\theta)=p(0)\) is independent
of \(\theta\). In other words, \(\gamma_0\) is the *constant
loop* at the point \(p(0)\in\mathbf{C}\setminus\{0\}\).

*(4.42)* When \(R\) is very large, the term \(z^n\) dominates in
\(p\), so \(\gamma_R(\theta)\approx \delta(\theta)\), where
\(\delta(\theta)=R^ne^{in\theta}\).

**Claim:** *(6.54)* There is a *homotopy invariant*
notion of *winding number around the origin* for paths in
\(\mathbf{C}\setminus\{0\}\) which gives zero for the constant loop
and \(n\) for the loop \(\delta(\theta)=R^ne^{in\theta}\).

*Homotopy invariant* means, roughly, invariant under continuous
deformations; in our situation, that means that the winding number
of \(\gamma_R\) around the origin should be independent of
\(R\). Since \(\gamma_0\) has winding number zero and \(\gamma_R\)
has winding number \(n\) for large \(R\), this implies \(n=0\).

## Outlook

*(9.24)* The rest of this module will be about defining this
notion of winding number, the notion of homotopy and homotopy
invariance, and generalising it to other spaces. In a more general
setting, the spaces we're interested in (in this example
\(\mathbf{C}\setminus\{0\}\)) will have an associated group (in this
example the integers \(\mathbf{Z}\)) called the *fundamental
group* and loops will have winding ``numbers'' which are elements
of this group.

This will have many applications, including:

- the Brouwer fixed point theorem (any continuous map from the 2-dimensional disc to itself has a fixed point).
- the fact that a trefoil knot cannot be unknotted.
- the fact that the three Borromean rings cannot be unlinked from one
another (despite the fact that they can be unlinked in pairs
ignoring the third).

# Pre-class questions

- Go through the rough sketch proof and highlight all the steps which
seem to you not to be fully justified. We will discuss this in
class, and I will call upon you for suggestions. Later in the
module, we will revisit this proof and fill in all the gaps
(hopefully to your satisfaction).

# Navigation

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**1.02 Paths, loops and homotopies**. - Index of all lectures.