21 The fundamental theorem of algebra
Let \(p(z) = \sum _{k=0}^n c_k z^k\) be a complex polynomial of degree \(n\ge 1\). If \(p(a)\ne 0\), then every disk \(D\) around \(a\) contains an interior point \(b\) with \(|p(b)| {\lt} |p(a)|\)
TODO
Every nonconstant polynomial with complex coefficients has at least one root in the field of complex numbers.
The rest is easy. Clearly, \(p(z)z^{-n}\) approaches the leading coefficient \(c_n\) of \(p(z)\) as \(|z|\) goes to infinity. Hence \(|p(z)|\) goes to infinity as well with \(|z| \to \infty \). Consequently, there exists \(R_1 {\gt} 0\) such that \(|p(z)| {\gt} |p(0)|\) for all points \(z\) on the circle \(\{ z : |z| = R_1 \} \). Furthermore, our third fact (C) tells us that in the compact set \(D_1 = \{ z : |z| \leq R_1 \} \) the continuous real-valued function \(|p(z)|\) attains the minimum value at some point \(z_0\). Because of \(|p(z)| {\gt} |p(0)|\) for \(z\) on the boundary of \(D_1\), \(z_0\) must lie in the interior. But by d’Alembert’s lemma 21.1 this minimum value \(|p(z_0)|\) must be \(0\) — and this is the whole proof.