Constructible field extension

A finite real extension $K/\mathbb{Q}$ is constructible when it is contained in a towerEquivalently, every element of $K$ can be obtained from rational numbers by field operations and successive square roots.

The real cyclotomic fieldis Galois of degree eight with cyclic Galois group $(\mathbb{Z}/17\mathbb{Z}{)}^{\times }/\{\pm 1\}$. A subgroup chain of successive index two gives a tower of three quadratic extensions, proving that $\cos (2\pi /17)$ is constructible.