Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 18J / b / Solution

A real subfield is constructible when all its elements are straightedge-and-compass constructible; algebraically, it is contained in a tower of quadratic extensions of $\mathbb{Q}$.

Let $\zeta =e^{2\pi i/17}$ andThe cyclotomic extension $\mathbb{Q}(\zeta )/\mathbb{Q}$ has Galois groupwhich is cyclic of order $16$. Complex conjugation corresponds to $-1$, so the maximal real subfield hasa cyclic group of order $8$.

This group has a chainin which every index is two. By the Galois correspondence, the fixed fields formwith every successive degree equal to two. Thus $F$, and in particular $\cos (2\pi /17)$, is constructible. This is the constructibility of the real seventeenth cyclotomic field.