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.