Are Splitting Fields Unique for Different Polynomial Families?

[PsychonautQQ]

So if E and E' are both extensions of K so that both E and E' are splitting fields of different families of polynomials in K[x], then E and E' are not isomorphic, correct? They need to be splitting fields for the same family of polynomials in K[x], correct?

 

[fresh_42]

So if E and E' are both extensions of K so that both E and E' are splitting fields of different families of polynomials in K[x], then E and E' are not isomorphic, correct? They need to be splitting fields for the same family of polynomials in K[x], correct?

I don't think so, at least not in this generality. They could still have the same zeros although the sets might be different. I think one has to consider the ideal generated by the two families.