In the discussion, it is established that if E and E' are both extensions of K serving as splitting fields for different families of polynomials in K[x], then E and E' are not isomorphic. The consensus is that splitting fields must correspond to the same family of polynomials in K[x] to maintain isomorphism. Additionally, the role of the ideal generated by the polynomial families is highlighted as a crucial factor in understanding the relationship between these fields.
PREREQUISITESThis discussion is beneficial for mathematicians, algebraists, and students studying field theory and polynomial algebra, particularly those interested in the relationships between splitting fields and polynomial families.
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.PsychonautQQ said: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?