Let ζ be a primitive 16th root of unity in $\mathbb{C}$.
(a) Compute the minimal polynomial of $ζ$ over $\mathbb{Q}$.
(b) Find the order and structure of the Galois group of $\mathbb{Q}(ζ) : \mathbb{Q}$.
I'm not sure how to do either part of this, but I am hoping if I have help with a) I can solve b) myself.
First note that $\zeta_{16}$ is a root of the polynomial $x^{16}-1$ so $\text{min}_{\zeta,\mathbb{Q}}$ divides $x^{16}-1.$ Now, we can factor $x^{16}-1$ in this way: $$x^{16}-1=(x^{8}-1)(x^{8}+1)=(x^{4}-1)(x^{4}+1)(x^8+1)=\cdots=(x-1)(x+1)(x^2+1)(x^4+1)(x^8+1).$$ Note now that $(x+1)^{2n}+1$ is irreducible by Eisenstein criterion (using $p=2$), so $\text{min}_{\zeta,\mathbb{Q}}$ is $x-1,x+1,x^2+1,x^4+1$ or $x^8+1.$ Clearly $x-1,x+1,x^2+1$ and $x^4+1$ do not have $\zeta_{16}$ as root, so $\text{min}_{\zeta,\mathbb{Q}}=x^8+1.$
For the other part: note that $\zeta_{16}^{k}$ is also a root of $x^{8}+1$ when $k$ is odd and $k\leq 15.$ Define $\sigma_{k}:\mathbb{Q}(\zeta)\rightarrow\mathbb{Q}(\zeta)$ by $\sigma_{k}(\zeta_{16})=\zeta_{16}^{k}.$
PS: In general, $x^{n}-1=\prod_{d|n}\Phi_{d},$ where $\Phi_{d}$ is the minimal polynomial of $\zeta_{d}$ (it is called "cyclotomic polynomial"). We also have that $[\mathbb{Q}(\zeta_{n}):\mathbb{Q}]=\varphi(n)$ (Euler function).