Solvable Brioschi resolvents (Q1396696)

This paper deals with the problem of determining whether a quintic polynomial is solvable, i.e. whether its roots are expressible by repeated radicals. Every irreducible quintic \(f(x)\) can be transformed, by two applications of Tschirnhausen transformations, into a so-called Brioschi quintic or Brioschi resolvent, with a parameter \(Z\) in the coefficients. The author derives a sextic rational function \(g(t)\) with the property that the parameter \(Z\) in the Brioschi resolvent satisfies \(Z=g(t)\) for some \(t\) if and only if the Brioschi resovlent and hence the given polynomial \(f\) is solvable.