Given that $f\left(x\right)$ is a polynomial, with rational coefficients.

Also, $\left(1+2i\right),\left( 2-\sqrt{3}\right), 5$ are roots of $f\left(x\right)$.

Since, imaginary and surd roots always exist in pairs when the coefficients are rational, hence $\left(1-2\mathrm{i}\right), \left(2+\sqrt{3}\right)$ are also roots of $f\left(x\right)=0$

$\therefore f\left(x\right)$ has $5$ at least roots.

Hence, the least value of '$n$' is $5$.