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 be a polynomial of degree . Since is a nonnegative continuous function, tending to as tends to , it has a minimum at some point :

, for all .

By division of by , write

where . Since is nonconstant, the integer is .
Therefore

, for all ,

and, expanding the square,

, for all .

Let , divide by , and let tend to . We obtain

,  for all real .

It is an easy exercise in algebra that, if is a complex number such that for all , then . Hence . Since , we obtain .