I hope somebody will help me with the following problem. Analytical solutions of cubic equations make use of the method of Cardano. Those solutions give roots that are functions of the coefficients of the equations, being functions where cubic roots are involved. Generally speaking cubic roots cannot be reduced to functions of quadratic roots, so there is no problem in that case. In a (also infinite) number of cases however the cubic roots can be written as quadratic roots or even more simple than that. Only in a few cases it is indeed possible to rewrite those cubic roots as functions of quadratic roots as in most of the cases the functions are too complex to find the functions with the quadratic roots. Now, it seems that there is a proof for the existence of an alternative solution for cubic equations, alternative in such a way that it gives either solutions with cubic roots when those cubic roots cannot be reduced to quadratic roots or rational numbers or it gives solutions without cubic roots. That means that Cardano actually generates cubic roots in the solutions. I was told that this conclusion is wrong, the argument being the theory of Galois. So, my question is: is there any theoretical argument that prohibits the existence of an alternative solution as mentioned before. I am very interested in an answer, as it can help me to save a lot of time to find a necessary parameter in the alternative solution