Solving solvable quintics

Abstract

to Cardano's formulas for the general cubic and quartic polynomials (cf. [1], Section 14.7, for example) and to determine the precise Galois group. In particular, the roots are produced in an order which is a cyclic permutation of the roots (see the Remark before the examples below), which can be useful in other computations (e.g. cf. [3]). We work over the rationals Q , but the results are valid over any field K of characteristic different from 2 and 5. Let x 1 ,x 2 ,x 3 ,x 4 ,x 5 be the roots of the general quintic polynomial x 5 - s 1 x 4 + s 2 x 3 - s 3 x 2 + s 4 x - s 5 where the s i are the eleme