A note on Solvable Equations including De Moivre’s quintic

Abstract

In this note we examine a simple algorithm for producing solvable polynomial equations and study the solutions of the quintic equation with real coefficients \[ x^{5} + 5ax^{3} + 5a^{2}x + b = 0\]
solved in radicals by De Moivre. the result is used to sum a hypergeometric series
for several arguments.