What are the properties of quartic polynomials with different cubic resolvents?

[Mystic998]

Homework Statement

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.

Homework Equations

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.

 

[lippyka]

So any polynomial would have to have a root in F but also have some other root that's not in F, or two distinct roots in F, but then the cubic resolvent would have to be different. Is it possible for two quartic polynomials to have the same splitting field but different cubic resolvents? I'm having trouble coming up with an example of this!