Editing Fundamental theorem of algebra (section)

===Geometric proofs===
There exists still another way to approach the fundamental theorem of algebra, due to J. M. Almira and A. Romero: by [[Riemannian geometry|Riemannian geometric]] arguments. The main idea here is to prove that the existence of a non-constant polynomial ''p''(''z'') without zeros implies the existence of a [[Flat manifold|flat Riemannian metric]] over the sphere '''S'''<sup>2</sup>. This leads to a contradiction since the sphere is not flat.

A Riemannian surface (''M'', ''g'') is said to be flat if its [[Gaussian curvature]], which we denote by ''K<sub>g</sub>'', is identically null. Now, the [[Gauss–Bonnet theorem]], when applied to the sphere '''S'''<sup>2</sup>, claims that

:<math>\int_{\mathbf{S}^2}K_g=4\pi,</math>

which proves that the sphere is not flat.

Let us now assume that ''n'' > 0 and

:<math>p(z) = a_0 + a_1 z + \cdots + a_n z^n \neq 0</math>

for each complex number ''z''. Let us define

:<math>p^*(z) = z^n p \left ( \tfrac{1}{z} \right ) = a_0 z^n + a_1 z^{n-1} + \cdots + a_n.</math>

Obviously, ''p*''(''z'')&nbsp;≠ 0 for all ''z'' in '''C'''. Consider the polynomial ''f''(''z'')&nbsp;=&nbsp;''p''(''z'')''p*''(''z''). Then ''f''(''z'')&nbsp;≠ 0 for each ''z'' in '''C'''. Furthermore,

:<math>f(\tfrac{1}{w}) = p \left (\tfrac{1}{w} \right )p^* \left (\tfrac{1}{w} \right ) = w^{-2n}p^*(w)p(w) = w^{-2n}f(w).</math>

We can use this functional equation to prove that ''g'', given by

:<math>g=\frac{1}{|f(w)|^{\frac{2}{n}}}\,|dw|^2 </math>

for ''w'' in '''C''', and

:<math>g=\frac{1}{\left |f\left (\tfrac{1}{w} \right ) \right |^{\frac{2}{n}}}\left |d\left (\tfrac{1}{w} \right ) \right |^2 </math>

for ''w''&nbsp;∈&nbsp;'''S'''<sup>2</sup>\{0}, is a well defined Riemannian metric over the sphere '''S'''<sup>2</sup> (which we identify with the extended complex plane '''C'''&nbsp;∪&nbsp;{∞}).

Now, a simple computation shows that

:<math>\forall w\in\mathbf{C}: \qquad  \frac{1}{|f(w)|^{\frac{1}{n}}} K_g=\frac{1}{n}\Delta \log|f(w)|=\frac{1}{n}\Delta \text{Re}(\log f(w))=0,</math>

since the real part of an analytic function is harmonic. This proves that ''K<sub>g</sub>''&nbsp;=&nbsp;0.