= 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 $\zeta=e^{2\pi i/17}$ and
$$
F=\mathbb Q(\zeta+\zeta^{-1})
=\mathbb Q\left(2\cos\frac{2\pi}{17}\right).
$$
The cyclotomic extension $\mathbb Q(\zeta)/\mathbb Q$ has Galois group
$$
(\mathbb Z/17\mathbb Z)^\times,
$$
which is cyclic of order $16$. Complex conjugation corresponds to $-1$, so the maximal real subfield has
$$
\operatorname{Gal}(F/\mathbb Q)
\cong(\mathbb Z/17\mathbb Z)^\times/\{\mathord\pm1\},
$$
a cyclic group of order $8$.
This group has a chain
$$
G=G_0\supset G_1\supset G_2\supset G_3=\{1\}
$$
in which every index is two. By the Galois correspondence, the fixed fields form
$$
\mathbb Q=F^{G_0}\subset F^{G_1}\subset F^{G_2}\subset F,
$$
with 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>.
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