Editing Fundamental theorem of algebra (section)

===Topological proofs===
[[File:Koreny.gif|thumb|right|Animation illustrating the proof on the polynomial <math>x^5-x-1</math>|150x150px]]

Suppose the minimum of |''p''(''z'')| on the whole complex plane is achieved at ''z''<sub>0</sub>; it was seen at the proof which uses Liouville's theorem that such a number must exist. We can write ''p''(''z'') as a polynomial in ''z''&nbsp;−&nbsp;''z''<sub>0</sub>: there is some natural number ''k'' and there are some complex numbers ''c<sub>k</sub>'', ''c''<sub>''k''&nbsp;+&nbsp;1</sub>, ..., ''c<sub>n</sub>'' such that ''c<sub>k</sub>''&nbsp;≠&nbsp;0 and:

:<math>p(z)=p(z_0)+c_k(z-z_0)^k+c_{k+1}(z-z_0)^{k+1}+ \cdots +c_n(z-z_0)^n.</math>

If ''p''(''z''<sub>0</sub>) is nonzero, it follows that if ''a'' is a ''k''<sup>th</sup> root of −''p''(''z''<sub>0</sub>)/''c<sub>k</sub>'' and if ''t'' is positive and sufficiently small, then |''p''(''z''<sub>0</sub>&nbsp;+&nbsp;''ta'')|&nbsp;<&nbsp;|''p''(''z''<sub>0</sub>)|, which is impossible, since |''p''(''z''<sub>0</sub>)| is the minimum of |''p''| on ''D''.

For another topological proof by contradiction, suppose that the polynomial ''p''(''z'') has no roots, and consequently is never equal to 0. Think of the polynomial as a map from the complex plane into the complex plane. It maps any circle |''z''|&nbsp;=&nbsp;''R'' into a closed loop, a curve ''P''(''R''). We will consider what happens to the [[winding number]] of ''P''(''R'') at the extremes when ''R'' is very large and when ''R'' = 0. When ''R'' is a sufficiently large number, then the leading term ''z<sup>n</sup>'' of ''p''(''z'') dominates all other terms combined; in other words,

:<math>\left | z^n \right | > \left | a_{n-1} z^{n-1} + \cdots + a_0 \right |.</math>

When ''z'' traverses the circle <math>Re^{i\theta}</math> once counter-clockwise <math>(0\leq \theta \leq 2\pi),</math> then <math>z^n=R^ne^{in\theta}</math> winds ''n'' times counter-clockwise <math>(0\leq \theta \leq 2\pi n)</math> around the origin (0,0), and ''P''(''R'') likewise. At the other extreme, with |''z''|&nbsp;=&nbsp;0, the curve ''P''(0) is merely the single point ''p''(0), which must be nonzero because ''p''(''z'') is never zero. Thus ''p''(0) must be distinct from the origin (0,0), which denotes 0 in the complex plane. The winding number of ''P''(0) around the origin (0,0) is thus 0. Now changing ''R'' continuously will [[homotopy|deform the loop continuously]]. At some ''R'' the winding number must change. But that can only happen if the curve ''P''(''R'') includes the origin (0,0) for some ''R''. But then for some ''z'' on that circle |''z''|&nbsp;=&nbsp;''R'' we have ''p''(''z'') = 0, contradicting our original assumption. Therefore, ''p''(''z'') has at least one zero.