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The fundamental theorem of algebra states that \mathbb{C} is algebraically closed, that is;

Theorem:

For any non-constant polynomial p in \mathbb{C}, there exists a z\in \mathbb{C} such that p(z)=0.

Proof:

Let B=(B_t: t \geq 0) be a Brownian motion on \mathbb{C} and suppose for a contradiction that a non-constant polynomial p does not have any zero’s. Let f:=1/p, then f is analytic and tends to 0 at infinity. Pick such that \alpha < \beta and note that \{Re f \leq \alpha\} and \{Re f \geq \beta\} contain an open set, which can be done due to the fact that f is continuous and non-constant.

Now f(B_t) is a continuous local martingale (by using Ito’s formula) and moreover it is bounded. Hence by the Martingale convergence we have that f(B_t) \rightarrow f(B)_\infty a.s. and in L^1.

This last statement is contradicted by the fact that Brownian motion is recurrent on the complex plane, in particular, it visits \{Re f \leq \alpha\} and \{Re f \geq \beta\} infinitely many times which gives that

\lim\inf f(B_t) \leq \alpha < \beta \leq \lim \sup f(B_t) a.s.

directly contradicting the Martingale convergence.

I found this little gem inĀ Rogers and Williams.

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