Let $\mathrm{p},\mathrm{q}\in \mathrm{ℝ}\text{ and }(1-\sqrt{3} i{)}^{200}=2^{199}(\mathrm{p}+\mathrm{iq}),\mathrm{i}=\sqrt{-1}$ Then p + q + q² and p - q + q² are roots of the equation.

$\begin{array}{l}(1+\sqrt{3}\mathrm{i}{)}^{200}=2^{199}(\mathrm{p}+\mathrm{iq})\\ 2^{200}{\left(\frac{1}{2}+\mathrm{i}.\frac{\sqrt{3}}{2}\right)}^{200}=2^{199}(\mathrm{p}+\mathrm{iq})\Rightarrow 2{\left(-{\mathrm{\omega }}^2\right)}^{200}=\mathrm{p}+\mathrm{iq}\\ \Rightarrow 2\omega =\mathrm{p}+\mathrm{iq}\Rightarrow \mathrm{p}=-1,\mathrm{q}=\sqrt{3}\end{array}$
$\begin{array}{r}\alpha =p+q+q^2=2-\sqrt{3}\\ \beta =p-q+q^2=2+\sqrt{3}\\ \alpha +\beta =4\\ \alpha \cdot \beta =1\end{array}$

Required equation is x² - 4x + 1 = 0