Fundamental Theorem of Algebra

Let p(z) = \sum_{j=0}^n a_j z^j where a_j and z\in\boldsymbol{{C}}, be a polynomial over the complex numbers. The fundamental theorem of algebra states there exist \alpha_j\in\boldsymbol{{C}} with p(z) = a_n\Pi_{j=0}^n (z - \alpha_j). The basic idea of the proof is p(z) behaves like a_n z^n as z gets large.

Lemma. If a polynomial p(z) is bounded then it is constant.

Proof: Recall p is bounded is there exists a constant M with |p(z)| < M for all z\in\boldsymbol{{C}}. If n=0 then p(z) = a_0 is constant. For n > 0 we have a_n z^n = p(z) + O(|z|^{n-1}) as |z|\to\infty so |a_n| R^n < M + CR^{n-1} for sufficiently large R. Dividing by R^n gives |a_n| < M/R^n + C/R which tends to 0 as R increases. This shows a_n = 0 for all n > 0 so p(z) = a_0 is constant.

Lemma. If p(\alpha) = 0 then p(z) = (z - \alpha)q(z) for some polynomial q(z).

Proof: By Taylor’s theorem p(z - z_0) = \sum_{j=0}^n p^{(j)}(z_0) (z - z_0)^j/j!. If p(z_0) = 0 then q(z) = \sum_{j=1}^n p^{(j)}(z - z_0)^{j-1}/j!.