Fundamental Theorem of Algebra/Proof 6

Theorem

Every non-constant polynomial with coefficients in $\C$ has a root in $\C$.

Let $p: \C \to \C$ be a complex, non-constant polynomial.

Aiming for a contradiction, suppose that $\map p z \ne 0$ for all $z \in \C$.

$\ds 0$

$<$

$\ds \cmod {\dfrac 1 {\map p 0} }$

$\ds $

$=$

$\ds \cmod {\frac 1 {2 \pi i} \oint_{\set {w \in \C : \cmod w = R} } \frac {\rd z} {z \cdot \map p z} }$

$\ds $

$\le$

$\ds \frac 1 {2 \pi} \max \limits_{\size z \mathop = R} \paren {\frac 1 {\size {z \cdot \map p z} } } 2 \pi R$

$\ds $

$=$

$\ds \max \limits_{\size z \mathop = R} \frac 1 {\size {\map p z} }$

$\ds $

$\to$

$\ds 0$

as $R \to +\infty$

$\blacksquare$