Quintics with Symmetries: Resolvents for Solvable Polynomials of Degree 5

It has been known for a long time that there is no general analytical solution to determine zeros of quintics (polynomials of degree 5) by radical expressions. However, certain classes of quintic equations can be solved, including both reducible and irreducible cases. In this book, a wide range of reducible quintics is analysed which can be linked to symmetric settings of zeroes. Specifically, if there are two square roots, or if more than two zeroes of a quintic equation are located in a circular symmetric fashion - be it centrally around 0 or an arbitrary point in the field of complex numbers - this corresponds, with the exception of a 5th roots scenario, to a reducible quintic. Solutions are provided by derived equations of lower degree than 5 which are presented in this book. In addition, various examples for quintic equations that fulfil the respective conditions are shown in detail.