General Formulas for Solving Solvable Sextic Equations

Let G be a transitive, solvable subgroup of S6. We show that there is a common formula for finding the roots of all irreducible sextic polynomials f(x) Q[x] with Gal(f) = G. Moreover, once the roots ri are calculated, there is an explicit procedure for numbering them so that the Galois group acts via τ(ri) = rτ(i) for τ G S6. We also demonstrate new criteria for determining the Galois group of an irreducible sextic polynomial f(x) Q[x]. The first two results generalize results of D. S. Dummit (1991, Math. Comp.57, 387–401) for quintic polynomials.