You can go 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.