So, I have to determine the Galois group of over , , .
That is irreducible over by Esienstein's criterion.
Resolvent is:
So is isomorphic to or .
To determine which, we check is irreducible over
Is this irreducible or not?
And what to do with , ?
Thank you!
Let be the splitting field for the resolvent, . As you have shown and so . Therefore, the Galois group is either or . It all depends whether or not is irreducible over . First has NO roots in because the roots of are: . These numbers have degree four over while is only a degree two extension. Thus, you need to show that is impossible for . To simplify things we will appeal to a result from number theory, that the ring of integers in is if . Thus, the ring of integers in is i.e. a field of fractions of is . By Gauss' Lemma (the general one) to prove irreduciblity of over it sufficies to prove irreducibility of over . Since we have where if we expand we get . I think (I did not check this!) that it is only possible to get if , now show that both those cases lead to impossibilities in the coefficiensts . Can you be the mathematician of the gaps and fill them in?