(This extends this post.) Define the function,
$$\sqrt[3]{G(t)} = \sqrt[3]{t+x_1}+\sqrt[3]{t+x_2}+\sqrt[3]{t+x_3}\tag1$$
where the $x_i$ are roots of the cubic,
$$x^3+ax^2+bx+c=0\tag2$$
While $G(t)$ generally is a root of a $9$th deg equation, davidoff303 found that, using $\cos(\frac{\pi\,k}{7})$ and special rational $t$, then $G(t)$ can in fact be rational. For example,
$$\sqrt[3]{\tfrac{74}{43}+2\cos\big(\tfrac{2\pi }{7}\big)}+\sqrt[3]{\tfrac{74}{43}+2\cos\big(\tfrac{4\pi }{7}\big)}+\sqrt[3]{\tfrac{74}{43}+2\cos\big(\tfrac{8\pi }{7}\big)}=2\,\sqrt[3]{\tfrac{49}{43}}$$
This post generalizes to $\cos(\frac{\pi\,k}{19})$, $\cos(\frac{\pi\,k}{37})$, etc. The trick is to use rational $t,w$ such that they obey,
$$t^3-at^2+bt-c=w^3\tag3$$
If $(3)$ holds, then there is the nice relation, $$Y(Y+d) = X^3\tag4$$
where $X,Y,$ and discriminant $d$ are,
$$27d^2 = 4 (a^2 - 3 b)^3 - (2 a^3 - 9 a b + 27 c)^2\tag5$$
$$X = b - 2 a t + 3 t^2 + (-a + 3 t) w + 3 w^2\tag6$$
$$Y =\tfrac{1}{2}(-a b - 9 c - d) + (a^2 t + 6 b t - 9 a t^2 + 9 t^3) + 3 (b - 2 a t + 3 t^2) w - 3 (a - 3 t) w^2\tag7$$
However, the product $Y(Y+d)$ by itself does not guarantee that $G(t)$ is rational.
Questions:
- If all three rational Diophantine eqns below, $$t^3-at^2+bt-c=w^3\\ Y = u^3\\ Y+d = v^3$$ are satisfied with $Y$ defined in $(7)$, then is $G(t)$ also rational?
- Let $(2)$ be the minimal cubic eqn of sums of $\cos(\frac{\pi\,k}{d})$ with $d=6n+1$. What is the necessary criteria for $d$ such that all three equations of Question 1 hold? (I observed only $d=7,19,37$ within the search bounds I used.)