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?

Not every field extension of degree $ 6 $ has a quadratic subfield. For example, $ \mathbf Q(\sqrt{1 + \sqrt[3]{2}}) $ has no subfield that is quadratic over $ \mathbf Q $. It is, however, true that every Galois extension of degree $ 6 $ has a unique quadratic subfield. It has been pointed out in another answer how to find this subfield of $ \mathbf Q(\zeta_7)/\mathbf Q $.
