Example in Abstract Algebra

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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.
 

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