Question

Let $1,w,w^2$ be the cube root of unity. The least possible degree of a polynomial with real coefficients having roots $2w,(2+3w),(2+3w^2),(2-w-w^2)$ is ____.

Answer 1

Since $1,\omega ,{\omega }^2$ are cube roots of unity, the following three properties hold:
a) ${\omega }^3=1$
b) ${\omega }^2+\omega =-1$
c) $\omega ,{\omega }^2$ are conjugates to each other
Since $\omega ,{\omega }^2$ are conjugates to each other $a+b\omega ,{a+b\omega }^2$ are also conjugates $\forall a,bϵR$ and $b\ne 0$ because $a$ is only added to real part after multiplying with $b$ which doesn't affect its properties.
Thus, $2+3\omega$ and ${2+3\omega }^2$ are conjugates.
Also, $2-\omega -{\omega }^2=3$ (substituting for $-{\omega }^2-\omega =1$)
Moreover, for any polynomial equation with real coefficients, complex roots exists in pairs. Since we already have a pair of complex roots and a real root, the conjugate of the root $2\omega$, which is $2{\omega }^2$, is sufficient to make all of them, the roots of a polynomial.
Thus, with the roots $2\omega ,2{\omega }^2,(2+3\omega ),(2+3{\omega }^2),(2-\omega -{\omega }^2)$, minimum degree of the required polynomial is $5$