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  • Dummit showed that given the five roots {$x_1, x_2, x_3, x_4, x_5$} of the quintic, then the expression,

$x_1^2(x_2 x_5+x_3 x_4)+x_2^2(x_1x_3+x_4x_5)+x_3^2(x_1x_5+x_2x_4)+x_4^2(x_1x_2+x_3 x_5)+x_5^2(x_1 x_4+x_2x_3)$

is the root of a $(5-2)!=3!=6$-deg eqn.

  • Boswell and Glasser gave an analogous one for the sextic {$y_1, y_2, y_3, y_4,y_5, y_6$} as,

$y_1^2y_2y_3+y_1y_2^2y_3+y_1y_2y_3^2+y_4^2y_5y_6+y_4y_5^2y_6+y_4y_5y_6^2$

which is the root of a 10th-deg eqn.

  • Question: What is a corresponding expression for the septic {$z_1, z_2, z_3, z_4,z_5, z_6, z_7$}?

(I know it will be a root of a $(7-2)! = 5! = 120$-deg equation, but I want to know what combination of the $z_i$ will be suitable.)

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    Hold on, how is the sextic resolvent you cite analogous to the quintic resolvent? By your reasoning it should have degree $(6 - 2)! = 24$, not $10$.2012-04-23
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    Boswell and Glasser's decic resolvent is analogous in the sense that if it has a rational root, then the sextic is solvable. Kindly see Theorem 2 of their paper at http://arxiv.org/pdf/math-ph/0504001v1.pdf. The formula $(p-2)!$ applies only to prime p.2012-04-23

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