Solving solvable quintics

Abstract

Abstract. Let f{x) = x +px +qx +rx + s be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if and only if f(x) is solvable by radicals (i.e., when its Galois group is contained in the Frobenius group F20 of order 20 in the symmetric group S5). When f(x) is solvable by radicals, formulas for the roots are given in terms of p, q, r, s which produce the roots in a cyclic order.

Standard Calculation of Fault Current Contribution of Doubly Fed Induction Generator-Based Wind Turbine

“…(17) is of fifth order, which can be solved in radicals, only if the sixth-degree resolvent possesses a rational solution (Galois Theory) [14]. Solving the quartic function as given in [15,16] would result in highly complex formula. However, if both G meas (s) and G dt (s) are neglected, considering that their time constants are in order of few hundreds of μs, the transfer function would be reduced to a quadratic function given by the following:…”

Commentary on an unpublished lecture by G. N. Watson on solving the quintic

“…In view of the current interest (both theoretical and computational) in solvable quintic equations [24], [33], [43]- [46], it seemed to the authors to be of interest to publish Professor Watson' [37] and at scattered places throughout his notebooks, especially in his first notebook [39]. Although many of Ramanujan's class invariants had been also calculated by Weber [57] [2], [3]; see also Berndt's book [1,Part V,Chapter 34].…”

“…It consequently became necessary for me to co-ordinate the work of previous writers in such a way as to have handy a systematic procedure for solving Abelian quintic equations as rapidly as possible, and this is what I am going to describe tonight. [24] and Kobayashi and Nakagawa [33]; see also [47].…”

“…Galois was the first mathematician to determine which quintic polynomials have roots expressible in terms of radicals, and in 1991 Dummit [24] …”

On the Lagrange resolvents of a dihedral quintic polynomial

“…By [1,Theorem 2] we know that the Galois group of f is Z 5 (cyclic group of order 5), D 5 (dihedral group of order 10), or F 20 (Frobenius group of order 20). When Gal(f ) D 5 , the splitting field K of f contains a unique quadratic subfield, say Q( √ m) (m squarefree integer ≠ 1).…”