I want to know how to calculate the degree of the field extension $[K:Q]$ where $K$ is the splitting field of $x^{13}+1$ over $\mathbb{Q}$.
I'm new to this area and this is not really covered in my course properly. So please don't assume I'm familiar to much when answering.
Since $-1$ is a root should I conclude that all roots are $-1w^{n}$, where $w\in\mathbb{C}$ and $w^{13}=1$ or am I searching for the solutions to $x^{13}=-1 \in\mathbb{C}$, or is this just the same thing since $-1\in\mathbb{Q}$ already?
How do I go about finding solutions to these equations in $\mathbb{C}$? After finding solutions how do I know which are the minimal polynomials satisfying these?
A lot of questions at the same time, but I don't really have anyone else to ask. Btw this is not a school assignment!
Are the roots $-w^{n}$ where $1\leq$n$\leq12$ and $w=\mathbb{e}^{\frac{2\pi}{13}i}$? And if we are searching for the $n$'th roots of unity when $n$ is composite why do we only include powers that are coprime to $n$?