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Quintic Equation

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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.

QuinticGaloisGroups

Irreducible quintic equations can be associated with a Galois group, which may be a symmetric group S_n, metacyclic group M_n, dihedral group D_n, alternating group A_n, or cyclic group C_n, 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

1024x^5-2816x^4+2816x^3-1232x^2+220x-11=0,
(1)

which arises in the computation of sin(pi/11).

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

a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0=0
(2)

into Bring quintic form

x^5-x+rho=0.
(3)

Defining

k=tan[1/4sin^(-1)((16)/(25sqrt(5)rho^2))]
(4)
s={-sgn(I[rho]) for R[rho]=0; sgn(R[rho]) for R[rho]!=0
(5)
b=(s(k^2)^(1/8))/(2·5^(3/4)sqrt(k(1-k^2))),
(6)

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

x_1=(-1)^(3/4)b{[m(e^(-2pii/5)q^(1/5))]^(1/8)+i[m(e^(2pii/5)q^(1/5))]^(1/8)}{[m(e^(-4pii/5)q^(1/5))]^(1/8)+[m(e^(4pii/5)q^(1/5))]^(1/8)}{[m(q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}
(7)
x_2=b{-[m(q^(1/5))]^(1/8)+e^(3pii/4)[m(e^(2pii/5)q^(1/5))]^(1/8)}×{e^(-3pii/4)[m(e^(-2pii/5)q^(1/5))]^(1/8)+i[m(e^(4pii/5)q^(1/5))]^(1/8)}{i[m(e^(-4pii/5)q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}
(8)
x_3=b{e^(-3pii/4)[m(e^(-2pii/5)q^(1/5))]^(1/8)-i[m(e^(-4pii/5)q^(1/5)]^(1/8))}×{-[m(q^(1/5))]^(1/8)-i[m(e^(4pii/5)q^(1/5))]^(1/8)}×{e^(3pii/4)[m(e^(2pii/5)q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}
(9)
x_4=b{[m(q^(1/5))]^(1/8)-i[m(e^(-4pii/5)q^(1/5))]^(1/8)}×{-e^(3pii/4)[m(e^(2pii/5)q^(1/5))]^(1/8)-i[m(e^(4pii/5)q^(1/5))]^(1/8)}{e^(-3pii/4)[m(e^(-2pii/5)q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}
(10)
x_5=b{[m(q^(1/5))]^(1/8)-e^(-3pii/4)[m(e^(-2pii/5)q^(1/5))]^(1/8)}×{-e^(3pii/4)[m(e^(2pii/5)q^(1/5))]^(1/8)+i[m(e^(-4pii/5)q^(1/5))]^(1/8)}{-i[m(e^(4pii/5)q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}.
(11)

where

m(q)=(theta_2^4(0,q))/(theta_3^4(0,q))
(12)

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

Euler reduced the general quintic to

x^5-10qx^2-p=0.
(13)

A quintic also can be algebraically reduced to principal quintic form

x^5+a_2x^2+a_1x+a_0=0.
(14)

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

x^5+ax+b=0
(15)

by showing that all irreducible solvable quintics with coefficients of x^4, x^3, and x^2 missing have the following form

x^5+(5mu^4(4nu+3))/(nu^2+1)x+(4mu^5(2nu+1)(4nu+3))/(nu^2+1)=0,
(16)

where mu and nu 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 epsilon=+/-1, c>=0, and e!=0 such that

a=(5e^4(3-4epsilonc))/(c^2+1)
(17)
b=(-4e^5(11epsilon+2c))/(c^2+1)
(18)

(Spearman and Williams 1994). The roots are then

x_j=e(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4),
(19)

where

u_1=((v_1^2v_3)/(D^2))^(1/5)
(20)
u_2=((v_3^2v_4)/(D^2))^(1/5)
(21)
u_3=((v_2^2v_1)/(D^2))^(1/5)
(22)
u_4=((v_4^2v_2)/(D^2))^(1/5)
(23)
v_1=sqrt(D)+sqrt(D-epsilonsqrt(D))
(24)
v_2=-sqrt(D)-sqrt(D+epsilonsqrt(D))
(25)
v_3=-sqrt(D)+sqrt(D+epsilonsqrt(D))
(26)
v_4=sqrt(D)-sqrt(D-epsilonsqrt(D))
(27)
D=c^2+1.
(28)

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

y^5+5ay^2+5by+c=0.
(29)

He then solved the related icosahedral equation

I(z,1,Z)=z^5(-1+11z^5+z^(10))^5 -[1+z^(30)-10005(z^(10)+z^(20))+522(-z^5+z^(25))]^2Z=0,
(30)

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

(Z^(-1/60)_2F_1(-1/(60),(29)/(60),4/5,1728Z))/(Z^(11/60)_2F_1((11)/(60),(41)/(60),6/5,1728Z)).
(31)

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

F_1(rho)=F_2(rho)
(32)
F_2(rho)=_4F_3(1/5,2/5,3/5,4/5;1/2,3/4,5/4;(3125)/(256)rho^4)
(33)
F_3(rho)=_4F_3(9/(20),(13)/(20),(17)/(20),(21)/(20);3/4,5/4,3/2;(3125)/(256)rho^4)
(34)
F_4(rho)=_4F_3(7/(10),9/(10),(11)/(10),(13)/(10);5/4,3/2,7/4;(3125)/(256)rho^4),
(35)
(36)

then the roots are

t_1=-rho_4F_3(1/5,2/5,3/5,4/5;1/2,3/4,5/4;(3125)/(256)rho^4)
(37)
t_2=-F_1(rho)+1/4rhoF_2(rho)+5/(32)rho^2F_3(rho)+5/(32)rho^3F_4(rho)
(38)
t_3=-F_1(rho)+1/4rhoF_2(rho)-5/(32)rho^2F_3(rho)+5/(32)rho^3F_4(rho)
(39)
t_4=-iF_1(rho)+1/4rhoF_2(rho)-5/(32)irho^2F_3(rho)-5/(32)rho^3F_4(rho)
(40)
t_5=iF_1(rho)+1/4rhoF_2(rho)+5/(32)irho^2F_3(rho)-5/(32)rho^3F_4(rho).
(41)

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

x^p+bx^q+c.
(42)

Consider the quintic

product_(j=0)^4[x-(omega^ju_1+omega^(4j)u_2)]=0,
(43)

where omega=e^(2pii/5) and u_1 and u_2 are complex numbers, which is related to de Moivre's quintic (Spearman and Williams 1994), and generalize it to

product_(j=0)^4[x-(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)]=0.
(44)

Expanding,

(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)^5-5U(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)^3-5V(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)^2+5W(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)+[5(X-Y)-Z]=0,
(45)

where

U=u_1u_4+u_2u_3
(46)
V=u_1u_2^2+u_2u_4^2+u_3u_1^2+u_4u_3^2
(47)
W=u_1^2u_4^2+u_2^2u_3^2-u_1^3u_2-u_2^3u_4-u_3^3u_1-u_4^3u_3-u_1u_2u_3u_4
(48)
X=u_1^3u_3u_4+u_2^3u_1u_3+u_3^3u_2u_4+u_4^3u_1u_2
(49)
Y=u_1u_3^2u_4^2+u_2u_1^2u_3^2+u_3u_2^2u_4^2+u_4u_1^2u_2^2
(50)
Z=u_1^5+u_2^5+u_3^5+u_4^5
(51)

(Spearman and Williams 1994). The u_is satisfy

u_1u_4+u_2u_3=0
(52)
u_1u_2^2+u_2u_4^2+u_3u_1^2+u_4u_3^2=0
(53)
u_1^2u_4^2+u_2^2u_3^2-u_1^3u_2-u_2^3u_4-u_3^3u_1-u_4^3u_3-u_1u_2u_3u_4
(54)
=1/5a
(55)
5[(u_1^3u_3u_4+u_2^3u_1u_3+u_3^3u_3u_4+u_4^3u_1u_2)-(u_1u_3^2u_4^2+u_2u_1^2u_3^2+u_3u_2^2u_4^2+u_4u_1^2u_2^2)]-(u_1^5+u_2^5+u_3^5+u_4^5)=b
(56)

(Spearman and Williams 1994).

See also

Bring Quintic Form, Brioschi Quintic Form, Bring-Jerrard Quintic Form, Cubic Equation, de Moivre's Quintic, Principal Quintic Form, Quadratic Equation, Quartic Equation, Sextic Equation

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References

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Quintic Equation

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Weisstein, Eric W. "Quintic Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/QuinticEquation.html

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