Least possible polynomial deegree of complex roots What is the least possible deegree of polynomial with real coefficients having roots $2\omega , 2+3\omega , 2+3\omega ^2 , 2-\omega -\omega ^2$
As there are four roots so the deegree should be four but the answer is given as five . how ?

Such a polynomial cannot have degree $4$, because we have, using $1+\omega+\omega^2=0$,
\begin{align}f(x) & =(x-2\omega)(x-(2+3\omega))(x-(2+3\omega^2))(x-(2-\omega-\omega^2))\\
 & =x^4 + 2x^3( - \omega - 2) + 2x^2(4\omega + 5) + x( - 20\omega - 21) + 42\omega,
\end{align}
which does not have real coefficients. Also, no scalar multiple of it has real coefficients.
