Fundamental Theorem of Algebra, Theorems of Brouwer and Borsuk Several months ago I was browsing through a question posted here ("Applications of Brouwer’s fixed point theorem"); amongst the comments attached it was mentioned that one could derive both Borsuk's Antipodal Theorem as well as the Fundamental Theorem of Algebra from Brouwer's Fixed-Point Theorem. However, to date I haven't seen (correct) proofs of these assertions. Could somebody kindly indicate where such proofs may be found ?
Thanks in advance,
St.
 A: As far as I'm aware, there isn't a compelling direct argument from Brouwer fixed point to imply the fundamental theorem of algebra.  Such an argument isn't impossible -- I can imagine some fairly contrived proofs but I don't know of a very natural one.  The references like Guillemin and Pollack don't derive FTOA from Brouwer, they derive both FTOA and Brouwer from degree/intersection theory.  In particular they only use mod-2 degree theory for Brouwer but oriented degree theory for FTOA.  
I had an argument written down here previously that I thought might work but now I realize it can't work.  Oh, but it's fixable. 
EDIT I've managed to repair the argument. The downside is it's not as simple.
A polynomial without roots produces a polynomial without fixed points.  Specifically, $p(z) \neq 0$ for all $z \in \mathbb C$ means $q(z) = p(z)+z$ has no fixed points in $\mathbb C$.  So what?  Think of $q(z)$ as a map of the Riemann sphere.  Now take the real oriented blow-up of the Riemann sphere at infinity (i.e. replace the point at infinity by its unit normal bundle in the sense of smooth real manifolds).  This is a disc.  So $q(z)$ becomes a smooth map of the disc, denote it $\hat q$, and identify the blow-up with $D^2$, the unit disc in $\mathbb C$ centred at the origin.
If $q(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_1 z + a_0$ and if we conjugate by $z \longmapsto 1/z$ so
$$\frac{1}{q(1/z)} = \frac{z^n}{1+a_{n-1}z + \cdots + a_0z^n}$$
So when you restrict $\hat q$ to the boundary circle, it becomes $z \longmapsto z^n$.
$z \longmapsto z^n$ has fixed points $z^{n-1}=1$, the $(n-1)$-th roots of unity.  So we can not directly appeal to Brouwer, since Brouwer's fixed point theorem might give you a pre-existing fixed point on the boundary.  
Consider the vector field $v(z) = z-\hat q(z)$ on $D^2$.  It is inward-pointing on the boundary circle with the sole exception of $z^{n-1}=1$, the $(n-1)$ roots of unity. But if we remove a small neighbourhood of $\partial D^2$ from $D^2$, the vector field $v$ restricts to an inward pointing vector field.  So you could appeal to Poincare-Hopf and say there has to be a zero in the interior, or you could talk about the flow of the vector field, and Brouwer's fixed-point theorem would then tell you the vector field must have a zero in the interior. 
So its not a slick proof, but it can be done. 
A suitable identification between $D^2$ and the blow-up of the Riemann sphere at infinity is done by the map $X : D^2 \to \hat{\mathbb C}$ given by $X(z) = \frac{1}{1-|z|^2} z$.  So $\hat q$ is the unique continuous extension of $X \circ q \circ X^{-1}$ to $D^2$. 
A: Since Ryan has discussed the relationship between Brouwer and FTOA, I would like to say something about Brouwer and Borsuk-Ulam（I prefer this name）. What I know is there are series of classical theorems on this direction, all closely related:
Thm.1(Brouwer) Continuous map $f:B^{n+1} \to B^{n+1}$ has at least one fixed point.
Thm.2(Hirsch) $\partial B^{n+1}$ is not a deformation retract of $B^{n+1}$.
Thm.3 $\mathrm{id}:S^n \to S^n$ is not homotopic to a constant map. 
Thm.4 Continuous map $g:S^n \to S^n$ sending antipodal points to antipodal points is not homotopic to a constant map.
Thm.5 There exists continuous map $h:S^n \to S^m$ sending antipodal points to antipodal points if and only if $n \leq m$.
Thm.6(Borsuk-Ulam)For continuous map $k:S^n \to \mathbb{R}^n$，there exists $x \in S^n$ such that $k(x)=k(-x)$.
Let me explain. Equivalence of 1 and 2 is quite standard, for example, see
Milnor  Topology from the differentiable viewpoint
By the way, it also contains a proof of FTOA, using the degree of maps.
To see the relation of 2 and 3, simply glue the boundary to a point. 
4 is an obvious generalization of 3. You may prove it by studying $H_*(\mathbb{R}P^n)$. This study also leads to a proof of 5.
Equivalence of 6 and (the "only if" part of) 5 is also not hard, you may take it as a exercise:)
All in all, what I have shown is some knowledge of $H_*(\mathbb{R}P^n)$ is sufficient for both Brouwer and Borsuk-Ulam. Hope this helps.
