I'm trying to determine the Galois groups of the following:
$x^8 -1$
$x^9 -1 $
For $x^8 -1$, I factored it, so I think we need to have $Q(i, {\sqrt 2})$ for the splitting field. We have 8 roots. We want to see the different ways to map them such that it's an automorphism, so for example we can check fixing one root and changing all the rest, or fixing two roots and changing all the rest etc. however, this seems very tedious. Is this the way this problem is meant to be solved, or am I missing some sort of trick?
This gets a bit worse for example if I'm trying to do this for $x^{12} -1$
In $x^9 -1$ I am having a problem seeing the splitting field. If someone can point me in the right direction, that would be great.