Galois group of quartic in field where every cubic has a root? I'm trying to figure out what the Galois group of an irreducible quartic whose discriminant is a square is, assuming that all cubics have a root. Since the discriminant is a square, the group is contained in $A_4$, and since the resolvent cubic has a root, this means the group is either $D_8$ or $V_4$. However, which one it is depends on whether the quadratic achieved by dividing the cubic by its root factor is reducible or not. Does every cubit having a root mean that this quadratic is also reducible? Or maybe that follows from the discriminant being a square? How to I choose between the two possible groups?

From the comments above:

Since $D_8$ is not contained in $A_4$, we see that the Galois group must be $A_4$.