I am trying to find the Galois group of the splitting field of $x^4 - 9x + 12$ over $\mathbb{Q}$. The polynomial is clearly irreducible (over $\mathbb{Q}$) by Eisenstein. In fact it has two pairs of complex conjugated roots (by computational tools), but I don't know the splitting field in more detail than that.
Obviously the Galois group is a subgroup of $S_4$, and I know that 4 divides the order of the Galois group so by Sylow there should be subgroups of order 4 and 2. A subgroup of order 2 means I have at least one transposition. The only other group element I am sure exists is $(12)(34)$, corresponding to complex conjugation, but it could be in any 4-subgroup. Judging by subgroups of $S_4$ I think the Galois group at this point is either $D_{2 \cdot 4}$ or $S_4$ (as it must be transitive, which I really only know because I read it in another thread).
I am struggling to reach further, as I can't think of any additional information telling me what group elements does and does not exist. The course hasn't gone into great detail on how to determine the Galois group for a certain polynomial, so I don't have very many tools available.
