Quintic Equations:

Bring-Jerrard Quintic Equations:

$\displaystyle x^{5}+p x+q=0$

$\displaystyle x^{5}\pm x\pm a=0$

Any general quintic can be reduced to Bring-Jerrard form using the Tschirnhaus type transformations:

$\displaystyle x^{5}+a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x+a_{0}=0$

1. Using the shifts ${ x+t}$ , we can remove the ${x^4}$ term.

2. It’s also possible to remove the ${x^3}$ term if we use the substitution ${y=x^2+ax+b}$ and choose ${a,b}$ so that ${y}$ satisfies ${y^5+ cy^2+dy+e=0.}$

Details:

$\displaystyle y^{5}+b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}=0$

where

$\displaystyle b_{4}=-a_4^{2}+2 a_3+a_4a-5 b$

$\displaystyle b_3=a_3^{2}-2 a_4 a_2+2 a_1-a_4 a_3 a+3 a_2 a+a_3 a^{2}+4 a_4^{2} b-8 a_3 b-4 a_4 a b+10 b^{2}$

We can make ${b_4=b_3=0}$ by these equations which give quadratic equations in ${a}$ or ${b}$ .

3. Bring-Jerrard Form: Using the subsituttion ${z =y^{4}+\alpha y^{3}+\beta y^{2}+\gamma y+\delta}$ , with the right choice of ${\alpha, \beta, \gamma, \delta}$ ,we can reduce the equation to

$\displaystyle z^5+pz+q=0$

Details:

$\displaystyle y^5+ cy^2+dy+e=0.$

$\displaystyle z^{5}+c_{1} z^{4}+c_{2} z^{3}+c_{3} z^{2}+c_{4} z+c_{5}=0$

$\displaystyle c_{1}=-5 \delta+3 \alpha c+4 d$

$\displaystyle \begin{array}{c} c_{2}=10 \delta^{2}-12 \alpha \delta c+3 \alpha^{2} c^{2}-3 \beta c^{2}+2 \beta^{2} d-16 \delta d+5 \alpha c d+6 d^{2}+5 \alpha \beta e \\ -4 c e+\gamma(3 \beta c+4 \alpha d+5 e) \end{array}$

It’s possible to solve for set ${c_1=c_2=c_3=0}$ solving at most cubic equations.

4. ${z =\frac{1}{\sqrt[4]{p}} y}$ reduces this to

$\displaystyle z^5+z-a=0$

5. Brioschi normal form:

Transformations of the form

$\displaystyle w=\frac{\lambda+\mu x}{\frac{x^{2}}{C}-3}$

reduce the general quintic to another one-parameter family given by

$\displaystyle w^{5}-10 C w^{3}+45 C^{2} w-C^{2}=0$

Bring-radical: We can solve the quintic in terms of Bring radicals- the solutions to ${x^5+x+a=0.}$

The Bring -radical ( ${a}$ ) has can be written in terms of hypergeometric functions and elliptic functions. In general, arbitrary polynomials can be solved in terms of elliptic integrals and modular functions, theta constants.

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