The polynomial equation of the lowest degree having roots 1, $\mathrm{i} \sqrt{3}$ is

$1,\sqrt{3}\mathrm{i},-\sqrt{3}\mathrm{i}$ are the roots then,

$$\begin{gathered} if \alpha ,\beta ,\gamma are roots then,the equation is \\ {x}^3-\left(\alpha +\beta +\gamma \right)x^2+\left(\alpha \beta +\beta \gamma +\gamma \alpha \right)x-\left(\alpha \beta \gamma \right)=0 \end{gathered}$$

$$\begin{gathered} x^3-\left(1+\cancel{\sqrt{3}i}-\cancel{\sqrt{3}i}\right)x^2+\left(\cancel{\sqrt{3}i}+3-\cancel{\sqrt{3}i}\right)x-\left(3\right)=0 \\ {x}^3-x^2+3x-3=0 \end{gathered}$$