However, any simple extension by a constructible point over the rationals must be a power of two, thus, for ten points, proving the impossibility of what straightedge-and-compass construction which baffled the ancient Greeks along with angle trisection and quadrature of the circle?
The polynomial "x cubed minus 2" is irreducible over the rational numbers and so its roots have degree three, as does a simple extension over the rationals by the real roots.  However, any simple extension by a constructible point over the rationals must be a power of two, thus, for ten points, proving the impossibility of what straightedge-and-compass construction which baffled the ancient Greeks along with angle trisection and quadrature of the circle?
The polynomial "x cubed minus 2" is irreducible over the rational numbers and so its roots have degree three, as does a simple extension over the rationals by the real roots.  
duplication (or doubling) of the cube or equivalents