I'm trying to pick up a little bit of Galois theory to see its applications to classical geometry and constructions of regular $n$-gons, the regular $17$-gon to be exact. This problem comes from Hartshorne's Classical Geometry and appears in Section 29 on Gauss's construction of a regular $17$-gon.
Let $\zeta=\cos(2\pi/7)+i\sin(2\pi/7)$ and let $\alpha=\zeta+\zeta^{-1}$.
(a) Find the minimal polynomial for $\alpha$ over $\mathbb{Q}$.
(b) Show that $\mathbb{Q}(\zeta)$ contains a unique subfield of $E$ of degree $2$ over $\mathbb{Q}$. Find an integer $d$ for which $E=\mathbb{Q}(\sqrt{d})$.
For $(a)$, I noticed that $\alpha=\zeta+\zeta^{-1}=\zeta+\zeta^6=2\cos(2\pi/7)$. Also, $ \alpha^2=\zeta^2+2+\zeta^5 $ $ \alpha^3=\zeta^3+\zeta^4+3\alpha $ and so $ \alpha^3+\alpha^2+\alpha=\zeta^6+\zeta^5+\zeta^4+\zeta^3+\zeta^2+\zeta+2+3\alpha $ and then $\alpha^3+\alpha^2-2\alpha=1$ since $\zeta$ is a root of $\Phi_7$, the seventh cyclotomic polynomial. So I believe the minimal polynomial of $\alpha$ is $x^3+x^2-2x-1$, which is irreducible by the rational roots test.
However for part (b), I'm coming up blank. I couldn't find an expression for $\zeta$ in terms of square roots, so I'm not really sure what $\mathbb{Q}(\zeta)$ looks like, nor can I hazard a guess as to what $d$ may be. I figured if $\zeta=\sqrt{10+2\sqrt{5}}$ or something, I could guess $d=5$ and attempt to work with that. Unfortunately, I'm not too keen with algebra, so my question is, how would one go about part (b)? Thanks.