Fundamental Theorem of Algebra/Proof 2

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

Proof
Let $p: \C \to \C$ be a complex polynomial with $p \left({z}\right) \ne 0$ for all $z \in \C$.

Then $p$ extends to a continuous transformation of the Riemann sphere $\hat{\C} = \C \cup \{\infty\}$ (and this extension also has no zeroes).

Since the Riemann sphere is compact, there is some $\varepsilon > 0$ such that $\left|{p \left({z}\right)}\right| \ge \varepsilon$ for all $z \in \C$.

Now consider the holomorphic function $g: \C \to \C$ defined by:
 * $g \left({z}\right) := \dfrac 1 {p \left({z}\right)}$

We have:
 * $\left|{g \left({z}\right)}\right| \le \frac 1 \varepsilon$

for all $z \in \C$.

By Liouville's Theorem, $g$ is constant.

Hence $p$ is also constant, as claimed.