In this e-book, a class of solvable quintics (polynomials of degree 5) is presented which, by specific modifications, are derived from quintics with 3rd roots. The latter ones represent one of the solvable classes of polynomials of degree 5 which are presented in the book Quintics with Symmetries (available as paperback version). In that book, resolvents for quintics are described which have a symmetric location of zeroes on a circle in the field of complex numbers. One of the simplest cases to imagine is a setting with the 5th unit roots as zeroes on the unit circle around zero, in the field of complex numbers, but many other scenarios can be thought of as well.
The idea is that there actually are various solvable quintics which
do not have zeroes in such a circular symmetry. Quintics with
3rd roots were chosen as a starting point for this
consideration because it seemed to be the most general scenario
which could be achieved by assuming changes of location of one or
two of the 3rd