Are the roots of $X^4+X^3-2$ constructible?

Question

As the title says, I'm trying to find out if the roots of the polynomial $f:=X^4+X^3-2\in\mathbb{C}[X]$ are constructible (from $\{0,1\}$).

I know that $X=1$ is a root of $f$ and $1$ is constructible. Furthermore I was able to factorise $f$ as $(X-1)(X^3+2X^2+2X+2)$, which reduces the question to finding out if the roots of the cubic $X^3+2X^2+2X+2$ are constructible.

I can't seem to progress any further this, but I think that the answer is that the roots of $X^3+2X^2+2X+2$ are not constructible, but I'm not completely sure about this.

Any help is greatly appreciated.

Answer 1

As noted, a root of $x^4+x^3-2=0$ is rational and certainly constructible, given of course a unit length segment for reference.

Setting aside that root, we have the cubic cequation $x^3+2x^2+2x+2=0$ for the remaining roots. This is easily proved not to have any more rational roots, so (in the case of a cubic equation) no more roots can be constructed with unmarked straightedge and compasses.

The solution of this particular cubic equation using a marked ruler figures into the marked ruler/compasses construction of the regular hendecagon[1].

Reference

1. Benjamin, Elliot & Snyder, Chip. (2014). "On the construction of the regular hendecagon by marked ruler and compass". Mathematical Proceedings of the Cambridge Philosophical Society. 156. 10.1017/S0305004113000753.