I am trying to compute the Galois group for the polynomial

, but I'm having a bit of trouble. I know that

over

. The polynomial will then split over

, which is an extension of degree 8.

Now I have the automorphisms

,

and

. I know that

. But I can't seem to get relations between them (and in particular, I would like to find an element of order 4 to show that this is the dihedral group). For example, I am interested in powers of

but I'm not sure how to run the computation.

Any suggestions would be much appreciated.