here, it may be helpful.
I'm trying to find the Galois Group of the splitting field of over Q.
This polynomial is irreducible, and the splitting field is generated by =a and w, any primitive 8th root of unity.
Here I am a bit uncertain. I believe like Q(a) is an extension of degree 8. And Q(w)=Q(i) is an extension of degree 2. So would Q(a,w) be an extension of degree 16. But this is the exact same as over Q, and that can't be correct.