It is shown that there is a common formula for finding the roots of all irreducible sextic polynomials \(f(X)\in \mathbb{Q}[X]\) with Galois group \(\text{Gal}(f)\) over \(\mathbb{Q}\) a transitive solvable subgroup of the symmetric group \(S_6\). The author gives an algorithm for the determination of the Galois group of an irreducible sextic polynomial over \(\mathbb{Q}\) and for determination of the roots by radicals, if it is solvable by radicals. Moreover, once the roots \(r_i\) are calculated, he gives a procedure for numbering them so that the Galois group acts via \(\tau(r_i)= r_{\tau(i)}\), for \(\tau\in \text{Gal}(f)\subset S_6\). There are sixteen non-isomorphic transitive subgroups of \(S_6\), up to conjugation, twelve of these groups are solvable and two of these are maximal solvable groups: \(G_{72}\) and \(G_{48}\).
The author explicitly computes in the Appendix the polynomials \(f_{10}\) and \(f_{15}\) of degree 10 and 15 that are Galois resolvents attached to the groups \(G_{72}\) and \(G_{48}\), and the discriminant of a general polynomial \(f(X)\in \mathbb{Q}[X]\) of degree 6. He gives explicit criteria for the determination of the Galois group of an irreducible polynomial \(f(X)\in \mathbb{Q}[X]\) of degree 6, in terms of factorization properties of \(f_{15}\) or \(f_{10}\) and to determine if the discriminant of \(f\) is a rational square. Explicit formulas for finding the roots of an irreducible, sextic polynomial \(f(X)\in \mathbb{Q}[X]\) with solvable Galois group are given in each of the three cases: \(\text{Gal}(f) \subset G_{72}\) and \(\text{Gal}(f) \nsubseteq G_{48}\); \(\text{Gal}(f) \subset G_{48}\), \(\text{Gal}(f) \nsubseteq G_{72}\); \(\text{Gal}(f) \subset G_{72}\cap G_{48}\). Two concrete examples are presented.
Results for quintic polynomials have been obtained by D. Dummit [Math. Comput. 57, 387-401 (1991; Zbl 0729.12008)].