Find coordinates for a regular heptagon in 3D Euclidean space where all $3$ components $(x,y,z)$ of all $7$ coordinates are elements of the same cubic field, or prove that it can't be done.
Background:
In his paper titled "Golden Fields: A Case for the Heptagon", mathematics professor Peter Steinbach describes the cubic field with basis $(1,\rho,\sigma)$, where $\rho$ and $\sigma$ are as follows:
$$\rho = 2\cos\left(\frac{\pi}7\right) = 1.8019377...$$
$$\sigma = 4\cos^2\left(\frac{\pi}7\right) - 1 = 2.2469796...$$
He describes relationships between the cubic field $\mathbb{Q}(1,\rho,\sigma)$ and the regular heptagon that are quite analogous to relationships between the quadratic field $\mathbb{Q}(\phi)$ and the regular pentagon, where $\phi = \frac{1}{2}(1+\sqrt 5)$ is the golden ratio.
In two dimensions, I believe it's not possible to find coordinates for a regular pentagon using numbers from $\mathbb{Q}(\phi)$, nor from any other quadratic field, but in three dimensions, it becomes possible, by using a tilted rotation axis like $(1,0,\phi)$. This fact is what allows an icosahedron, dodecahedron, and many other polyhedra with icosahedral rotational symmetry to be represented using numbers from $\mathbb{Q}(\phi)$.
Similarly, I think there is no 2D solution for the regular heptagon using numbers from $\mathbb{Q}(1,\rho,\sigma)$ nor from any other cubic field, which leads me to the question of whether it's possible in 3D.
