Galois Theory Quadratic Subfield Let $ζ_7=e^{i2\pi/7}$ be a 7th root of unity. The field $\Bbb{Q}(ζ_7)$ contains a quadratic subfield that can be expressed in the form of $\Bbb{Q}(\sqrt{D})$ where D is an integer. What is D? 
I understand that there is a field extension of order 6 and therefore there will be a quadratic subfield, but how do we find out what D is?

Let $\displaystyle\alpha = \sum_{n=0}^{p-1} \left(\frac{n}{p}\right) \zeta_p^n$ be the Gauss sum, then $\alpha=  \sqrt{-p}$ if $p \equiv 3 \bmod 4$ and $\mathbb{Q}(\alpha)= \mathbb{Q}(\sqrt{-p})$ is the quadratic subfield of $\mathbb{Q}(\zeta_p)$.
$\text{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q}) \simeq \mathbb{Z}_p^\times$. Let $H$ be its subgroup of order $\frac{p-1}{2}$. It corresponds to the quadratic residues in $\mathbb{Z}_p^\times$. Then $\mathbb{Q}(\alpha)$ is the fixed field of $H$.
