Example in Abstract Algebra
1. The problem statement, all variables and given/known data
I'm trying to come up with an example of a quartic polynomial over a field F which has a root in F, but whose splitting field isn't the same as its resolvent cubic.
2. Relevant equations
3. The attempt at a solution
Well, I know the splitting field of the cubic resolvent is contained in the splitting field of the quartic, so that narrows down the choices quite a bit. But frankly I'm not even sure this is possible because the cubic resolvent is pretty much defined in terms of the roots of the quartic.