My question is related to this previous one. I was wondering what is the minimal number of steps $S(a)$ to construct a number $a \in \mathbb R$ that is constructible (as defined here).
For instance, I could construct $\sqrt 2$ in 6 steps (is it possible to do better ?). By "step", I mean either drawing a circle when the compass is placed, or drawing a line when the straightedge is placed.
Following the ideas found on this page (see "Best Known Constructions"), here is a construction of $\cos(2\pi / 5)$ in 10 steps (if $|P_1P_2| = 1$, then this is the red segment $P_2A$) :
Is it possible to do better ? What are some interesting bounds on the minimal number of steps to construct $x_n = \cos(2 \pi / n)$, with $n=2^kF_1 \cdots F_r$ and $F_i$ are distinct Fermat primes ?
Thank you very much !