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The fundamental theorem of algebra states that is algebraically closed, that is;

Theorem:

*For any non-constant polynomial in , there exists a such that .*

Proof:

Let be a Brownian motion on and suppose for a contradiction that a non-constant polynomial does not have any zero’s. Let , then is analytic and tends to 0 at infinity. Pick such that and note that and contain an open set, which can be done due to the fact that is continuous and non-constant.

Now is a continuous local martingale (by using Ito’s formula) and moreover it is bounded. Hence by the Martingale convergence we have that a.s. and in .

This last statement is contradicted by the fact that Brownian motion is recurrent on the complex plane, in particular, it visits and infinitely many times which gives that

a.s.

directly contradicting the Martingale convergence.

I found this little gem inĀ Rogers and Williams.