Quintics with One or Two 3rd Roots

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.



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 root zeroes. It was found that, if certain symmetry patterns are given, the resulting quintics can be resolved. The approach, derivations and examples are presented in this text.