Squares in a triangular lattice

Question

It is not difficult to show that the points of an integer lattice do not span any equilateral triangles (for example, see here).

Is it also true that the points of a triangular lattice do not span any squares?

That is, a lattice that is composed of the vertices of a tiling of equilateral triangles, or equivalently: $\left\{a\cdot (1,0) + b\cdot (1/2, \sqrt{3}/2) \ \mid \mbox{ for any } a,b\in{\mathbb Z} \right\}$.

Thanks,

Adam

Answer 1

With $\rho=\frac{1+i\sqrt 3}2$, we have $\rho\notin\mathbb Q[i]$ and $i\notin\mathbb Q[\rho]$