Quintic Equation

Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions, as rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois. However, certain classes of quintic equations can be solved in this manner.

Irreducible quintic equations can be associated with a Galois group, which may be a symmetric group , metacyclic group , dihedral group , alternating group , or cyclic group , as illustrated above. Solvability of a quintic is then predicated by its corresponding group being a solvable group. An example of a quintic equation with solvable cyclic group is

which arises in the computation of .

In the case of a solvable quintic, the roots can be found using the formulas found in 1771 by Malfatti, who was the first to "solve" the quintic using a resolvent of sixth degree (Pierpont 1895).

The general quintic can be solved in terms of Jacobi theta functions, as was first done by Hermite in 1858. Kronecker subsequently obtained the same solution more simply, and Brioschi also derived the equation. To do so, reduce the general quintic

into Bring quintic form

Defining

where is the elliptic modulus, the roots of the original quintic are then given by

where

is the inverse nome, which is expressible as a ratio of Jacobi theta functions.

Euler reduced the general quintic to

A quintic also can be algebraically reduced to principal quintic form

By solving a quartic, a quintic can be algebraically reduced to the Bring quintic form, as was first done by Jerrard. Runge (1885) and Cadenhad and Young found a parameterization of solvable quintics in the form

by showing that all irreducible solvable quintics with coefficients of , , and missing have the following form

where and are rational.

Spearman and Williams (1994) showed that an irreducible quintic of the form (15) having rational coefficients is solvable by radicals iff there exist rational numbers , , and such that

(Spearman and Williams 1994). The roots are then

where

Felix Klein used a Tschirnhausen transformation to reduce the general quintic to the form

He then solved the related icosahedral equation

where is a function of radicals of , , and . The solution of this equation can be given in terms of hypergeometric functions as

Another possible approach uses a series expansion, which gives one root (the first one in the list below) of the Bring quintic form. All five roots can be derived using differential equations (Cockle 1860, Harley 1862). Let

then the roots are

This technique gives closed form solutions in terms of hypergeometric functions in one variable for any polynomial equation which can be written in the form

Consider the quintic

where and and are complex numbers, which is related to de Moivre's quintic (Spearman and Williams 1994), and generalize it to

Expanding,

where

(Spearman and Williams 1994). The s satisfy

(Spearman and Williams 1994).