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Brioschi Quintic Form

Using a Tschirnhausen transformation, the principal quintic form can be transformed to the one-parameter form

w^5-10cw^3+45c^2w-c^2=0
(1)

named after Francesco Brioschi (1824-1897) and which is important to the Klein's solution of the general quintic in terms of hypergeometric functions (Doyle and McMullen). This can be attained by using the transformation,

x=(aw+b)/(c^(-1)w^2-3)
(2)

(Dickson 1959) and eliminating the variable w between the two using resultants to form a new quintic

x^5+f_1x^2+f_2x+f_3=0,
(3)

where

f_1=5c(8b^3+ab^2+a^3c+72a^2bc)/(1728c-1)
(4)
f_2=5c(-b^4+a^3bc+18a^2b^2c+27a^4c^2)/(1728c-1)
(5)
f_3=c(b^5-10a^2b^3c+a^5c^2+45a^4bc^2)/(1728c-1).
(6)

Equating coefficients with a generic principal quintic

x^5+5rx^2+5sx+t=0
(7)

results in a system of three equations in the three unknowns a, b, and c. Amazingly, this can be resolved to a single equation that is only a quadratic and given in the variable b by

(r^4-s^3+rst)b^2+(-11r^3s-2s^2t+rt^2)b+(64r^2s^2-27r^3t-st^2)=0
(8)

(Dickson 1959).

See also

Principal Quintic Form, Quintic Equation, Tschirnhausen Transformation

Portions of this entry contributed by Tito Piezas III

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References

Dickson, L. Algebraic Theories. New York: Dover, pp. 245-247, 1959.Doyle, P. and McMullen, C. "Solving The Quintic By Iteration." http://abel.math.harvard.edu/_ctm/papers/home/text/papers/icos/icos.pdf.Piezas, T. "Deriving the Bring-Jerrard Quintic Using a Quadratic Transformation." http://www.geocities.com/titus_piezas/Brioschi.html.

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Brioschi Quintic Form

Cite this as:

Piezas, Tito III and Weisstein, Eric W. "Brioschi Quintic Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrioschiQuinticForm.html

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