<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] Let's look at the given roots. We see that one of these roots,
$2-\omega -\omega ^2$, is in fact real:
$$
2-\omega -\omega ^2 = 3, \quad\mbox{ while }\quad
(x-(2+3\omega))(x-(2+3\omega^2)) = x^2-x+7.
$$
A fourth degree polynomial with the four given roots would have complex coefficients, namely, it's the polynomial
$$
P_4(x) =  (x - 3) ( x - 2\omega ) (x^2 - x + 7). \tag{1}
$$
There exists a $5$th-degree polynomial that has all four given roots and an additional fifth root:
$$
P_5(x) = (x - 3) ( x - 2\omega )( x - 2\bar\omega ) (x^2 - x + 7)
$$
$$
 = x^5 - 2 x^4 + 6 x^3 - 17 x^2 - 2 x - 84. \tag{2}
$$
So the sought-for minimal degree is $5$; the polynomial $(2)$ proves this.
The additional root, $2\bar\omega$, is obtained by complex conjugation of the root $2\omega$ of $P_4(x)$, eq. $(1)$.
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