Constructibility of the real seventeenth cyclotomic field
= Constructibility of the real seventeenth cyclotomic field
The real cyclotomic field
$$
\mathbb Q(\zeta_{17}+\zeta_{17}^{-1})
$$
is Galois of degree eight with cyclic Galois group $(\mathbb Z/17\mathbb Z)^\times/\{\mathord\pm1\}$. A subgroup chain of successive index two gives a tower of three quadratic extensions, proving that $\cos(2\pi/17)$ is constructible.