Solvable Brioschi Resolvents

Abstract

The Brioschi resolvent y 5 −10Zy 3 +45Z 2 y−Z 2 is a key component of an algorithm for calculating the roots of a general quintic polynomial. It is obtained from the general quintic polynomial by applying two Tschirnhausen transformations. In this paper it is shown that if the quintic polynomial is a solvable polynomial, then its associated parameter Z in the Brioschi resolvent satisfies Z=g(t) where g(t) is a rational function in ℚ(t) and t is chosen from an appropriate field.