## 3 April 2011

### A new very short proof of the fundamental theorem of algebra

I've always been intrigued by the fundamental theorem of algebra (every nonconstant polynomial with complex coefficients has a root), not least because I don't know any proof which uses algebra only. Earlier, I posted an easy proof in this blog, one that uses Cauchy's theorem.There is a recent proof (Oswaldo Rio Branco de Oliveira, Mathem. Intellig., March 2011), which is almost trivial. It goes as follows (and this is a chance for me to see if the embedded LaTeX script works...):

Let $P(z)$ be a polynomial of degree $n$. Since $|P(z)|$ is a nonnegative continuous function, tending to $\infty$ as $|z|$ tends to $\infty$, it has a minimum at some point $z_0$:
$|P(z)| \ge |P(z_0)|$, for all $z$.
By division of $P(z)-P(z_0)$ by $z-z_0$, write
$P(z) = P(z_0) + (z-z_0)^k Q(z-z_0),$
where $Q(0) \not = 0$. Since $P(z)$ is nonconstant, the integer $k$ is $\ge 1$.
Therefore
$|P(z_0) + (z-z_0)^k Q(z-z_0)|^2 \ge |P(z_0)|^2$, for all $z$,
and, expanding the square,
$|z-z_0|^{2k} |Q(z-z_0)|^2 + 2 \Re \{ (z-z_0)^k Q(z-z_0) \overline{P(z_0)}\} \ge 0$, for all $z$.
Let $z=z_0 + r e^{i \theta}$, divide by $r^k$, and let $r$ tend to $0$. We obtain
$\Re \{ e^{i k \theta} Q(0) \overline{P(z_0)}\} \ge 0$,  for all real $\theta$.
It is an easy exercise in algebra that, if $\alpha$ is a complex number such that $\Re \{ e^{i k \theta} \alpha\} \ge 0$ for all $\theta$, then $\alpha=0$. Hence $Q(0) \overline{P(z_0)}=0$. Since $Q(0) \neq 0$, we obtain $P(z_0)=0$.

T H E B O T T O M L I N E

## What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

## The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

## The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant

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