I'm struggling with a question I've got as I can't seem to wrap my head around Galois theory much.
i)Prove that $\zeta^{2}$ is a primitive 8th root of unity and $\zeta^{4}$ is a primitive 4th root of unity. Deduce then that $\zeta$ is a constructible number, where $\zeta$ is primitive 16th root of unity
For this one I'm unsure again, since the only question I've done like it, is when showing that $-\zeta$ is the primitive 5th root of unity when $\zeta$ is the primitive 10th root of unity but I'm confused on how I'm meant to apply that knowledge here.
ii) Find the order and structure of the Galois group of $\mathbb{Q}(\zeta):\mathbb{Q}$, where $\zeta$ is the primitive 16th root of unity in $\mathbb{C}$.
I found this Structure of $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$? which I think may be what I'm looking for in terms of helping my understanding and applying it to $\zeta_{16}$ but I'm unsure.
Any help would be appreciated!