<s>[INST] 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 ?
 [/INST] The least possible degree of polynomial with real coefficients having these roots is the product of $(x-\alpha)(x-\bar\alpha)$, where $\alpha$ are the given roots, unless $\alpha$ is real, in which case the second factor is omitted.
Since $\bar\omega=\omega^2$ and $1+\omega+\omega^2=0$, the polynomial is
$$
(x-2\omega)(x-2\bar\omega)(x-2-3\omega)(x-2-3\bar\omega)(x-3)
\\=
(x^2+2x+4)(x^2-x+7)(x-3)
\\=
x^5 - 2 x^4 + 6 x^3 - 17 x^2 - 2 x - 84
$$
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