Fundamental theorem of algebra

Every nonconstant complex polynomial has a complex root and therefore factors completely into linear factors over $\mathbb{C}$.

If a nonconstant complex polynomial were zero-free, its reciprocal would be an entire bounded function. The real part of its value along planar Brownian motion is a bounded martingale, hence converges almost surely. If that real part were nonconstant, continuity would give two disks with separated value ranges, and recurrence of planar Brownian motion would force alternate visits to both at arbitrarily large times, contradicting convergence. It is therefore constant; the Cauchy-Riemann equations then make the reciprocal constant, a contradiction.

If a nonconstant complex polynomial $P$ had no zeros, its reciprocal $g=1/P$ would be an entire holomorphic function, bounded and tending to zero at infinity. Its real and imaginary parts are harmonic functions. Thus $g(z+B_t)$, for planar Brownian motion started at zero, is a bounded complex martingale with expectation $g(z)$. For fixed $R$, the Gaussian density bounds $\mathbb{P}(|z+B_t|\le R)$ by $R^2/(2t)$. Outside a sufficiently large disk, $|g|\le \epsilon$. Hence $|g(z)|\le \epsilon +\Vert g{\Vert }_{\infty }{R}^2/(2t)$. Let $t\to \infty$ and then $\epsilon \downarrow 0$ to obtain $g(z)=0$, a contradiction. This proves the fundamental theorem of algebra without the maximum modulus principle.