Algebra/Chapter 10/Fundamental Theorem

‘For every non-constant polynomial with complex coefficients, there exists at least one complex root.

Furthermore, its degree is also the amount of its roots (with multiplicity).

Proof

Let there be a non-constant polynomial

$p(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{1}z+a_{0}\quad (a_{n}\neq 0)$

Then we have $\lim _{z\to \infty }|p(z)|=\infty$. Since the function $|p(z)|$ is continuous, there exists a $z_{0}\in \mathbb {C}$ such that $|p(z_{0})|=\min _{z\,\in \,\mathbb {C} }|p(z)|$.

Let us write $p(z)=p(z_{0})+(z-z_{0})^{m}q(z)$, for $m\in \mathbb {N}$ and a polynomial $q(z)$ such that $q(z_{0})\neq 0$.

Let ${\overline {p(z_{0})}}$ be the complex conjugate of $p(z_{0})$. Then for all $z\in \mathbb {C}$ we get:

${\begin{aligned}&|p(z_{0})|^{2}\leq |p(z)|^{2}=|p(z_{0})|^{2}+|z-z_{0}|^{2m}|q(z)|^{2}+2\,{\text{Re}}\!\left[\,{\overline {p(z_{0})}}(z-z_{0})^{m}q(z)\,\right]\\[5pt]&|z-z_{0}|^{2m}|q(z)|^{2}+2\,{\text{Re}}\!\left[\,{\overline {p(z_{0})}}(z-z_{0})^{m}q(z)\,\right]\geq 0\end{aligned}}$

Let $z=z_{0}+re^{\theta i}$ for $r>0$:

${\begin{aligned}&r^{2m}{\bigl |}q(z_{0}+re^{\theta i}){\bigr |}^{2}+2\,{\text{Re}}\!\left[\,{\overline {p(z_{0})}}\,r^{m}e^{m\theta i}q(z_{0}+re^{\theta i})\,\right]\geq 0\\[5pt]&r^{m}{\bigl |}q(z_{0}+re^{\theta i}){\bigr |}^{2}+2\,{\text{Re}}\!\left[\,{\overline {p(z_{0})}}q(z_{0}+re^{\theta i})e^{m\theta i}\,\right]\geq 0\end{aligned}}$

Taking the limit as $r\to 0^{+}$ yields:

${\text{Re}}\!\left[\,{\overline {p(z_{0})}}q(z_{0})e^{m\theta i}\,\right]\geq 0$

Let $c={\overline {p(z_{0})}}q(z_{0})$ and $\omega ={\text{cis}}\!\left({\frac {\pi }{2m}}\right)$.

By plugging $e^{\theta i}=1,\omega ,\omega ^{2},\omega ^{3}$ into the inequality and by de Moivre's formula, we get:

${\begin{aligned}&{\text{Re}}(\pm c)=\pm \,{\text{Re}}(c)\geq 0\\&{\text{Re}}(\pm ci)=\mp \,{\text{Im}}(c)\geq 0\end{aligned}}$

hence ${\text{Re}}(c)={\text{Im}}(c)=0$ and so ${\overline {p(z_{0})}}q(z_{0})=0$.

Therefore, from $q(z_{0})\neq 0$ we get ${\overline {p(z_{0})}}=p(z_{0})=0$.

$\blacksquare$