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.
I'm interested in knowing how did you come up with those three automorphisms, as you didn't define them
over a basis of the splitting field over the rationals...
I mean: (1) How can you actually know the three automorphisms (i.e., how can you tell they're even different, to
begin with)?, and (2) How can you possibly find out any relation between them if you don't know the maps?