Since 1,ω,ω2 are cube roots of unity, the following three properties hold:
a) ω3=1
b) ω2+ω=−1
c) ω,ω2 are conjugates to each other
Since ω,ω2 are conjugates to each other a+bω,a+bω2 are also conjugates ∀a,bϵR and b≠0 because a is only added to real part after multiplying with b which doesn't affect its properties.
Thus, 2+3ω and 2+3ω2 are conjugates.
Also, 2−ω−ω2=3 (substituting for −ω2−ω=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 2ω, which is 2ω2, is sufficient to make all of them, the roots of a polynomial.
Thus, with the roots 2ω,2ω2,(2+3ω),(2+3ω2),(2−ω−ω2), minimum degree of the required polynomial is 5