Out[1335]= -5 - 2 x + 7 x^2 - 4 x^3 + x^4 + x^6>>>>>> In[1337]:= Solve[m^3 - 4 m - 2 == 0]>>>> Out[1337]= {{m -> (9 + I Sqrt[111])^(1/3)/3^(2/3) +>> 4/(3 (9 + I Sqrt[111]))^(>> 1/3)}, {m -> -(((1 + I Sqrt[3]) (9 + I Sqrt[111])^(1/3))/(>> 2 3^(2/3))) - (2 (1 - I Sqrt[3]))/(3 (9 + I Sqrt[111]))^(>> 1/3)}, {m -> -(((1 - I Sqrt[3]) (9 + I Sqrt[111])^(1/3))/(>> 2 3^(2/3))) - (2 (1 + I Sqrt[3]))/(3 (9 + I Sqrt[111]))^(1/3)}}>>>> In[1338]:= Factor[%1335, Extension -> m /. %[[1]]]>>>> Out[1338]= -(1/>> 27) (15 + (48 3^(1/3))/(9 + I Sqrt[111])^(>> 2/3) + (3 (9 + I Sqrt[111]))^(>> 2/3) + ((12 3^(2/3))/(9 + I Sqrt[111])^(1/3) +>> 3 (3 (9 + I Sqrt[111]))^(1/3)) x + 9 x^2) (-3 + (>> 48 3^(1/3))/(9 + I Sqrt[111])^(2/3) - (>> 8 3^(2/3))/(9 + I Sqrt[111])^(1/3) ->> 2 (3 (9 + I Sqrt[111]))^(1/3) + (3 (9 + I Sqrt[111]))^(>> 2/3) + (6 - (4 3^(2/3))/(9 + I Sqrt[111])^(>> 1/3) - (3 (9 + I Sqrt[111]))^(1/3)) x ->> 6 x^2 + ((4 3^(2/3))/(9 + I Sqrt[111])^(>> 1/3) + (3 (9 + I Sqrt[111]))^(1/3)) x^3 - 3 x^4)
It would be *really* nice to find a simple solver for this sqrt(cuberts) case of the sextic, which I think would finish the problem, with the surprising result that sextic solutions, instead of involving an enormous pile of 5th, 4th, 3rd, ... roots, are, if anything, simpler than quartic solutions! (And infinitely rarer.) The inverse Fourier hack *does* work for the septic below and its resolvent sextic, giving a large expression with 7th roots(cuberts(sqrts)) vs 29th roots of 1. Applying this expression to any 7 consecutive integers gives all 7 roots: (1/(2^(2/7)*7^(6/7)))*((-((29*(856512704956195*2^(2/3) - 856512704956195*I*2^(2/3)*Sqrt[3] + 236175961274505*I*2^(2/3)*Sqrt[7] + 236175961274505*2^(2/3)*Sqrt[21] + 82410846*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) + 22757210*I*Sqrt[7]*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) + 2^(1/3)*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3) + I*2^(1/3)*Sqrt[3]*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3)))/(3*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3))))^(1/7)* E^((2/7)*I*Pi*(4 - 3*#1))) + (1/(2^(2/7)*7^(6/7)))* (((29*(-856512704956195*2^(2/3) + 856512704956195*I*2^(2/3)*Sqrt[3] + 236175961274505*I*2^(2/3)*Sqrt[7] + 236175961274505*2^(2/3)*Sqrt[21] - 82410846*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) + 22757210*I*Sqrt[7]* (-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) - 2^(1/3)*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3) - I*2^(1/3)*Sqrt[3]*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3)))/(3*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3)))^(1/7)* E^((2/7)*I*Pi*(3 - 2*#1))) + (1/7^(6/7))*(((29*(856512704956195*2^(2/3) + 236175961274505*I*2^(2/3)*Sqrt[7] - 41205423*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) - 11378605*I*Sqrt[7]*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) + 2^(1/3)*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3)))/ (6*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3)))^(1/7)*E^((2*I*Pi*#1)/7)) + (1/7^(6/7))*(((29*(856512704956195*2^(2/3) - 236175961274505*I*2^(2/3)*Sqrt[7] - 41205423*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) + 11378605*I*Sqrt[7]*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) + 2^(1/3)*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3)))/ (6*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3)))^(1/7)*E^((12*I*Pi*#1)/7)) + (1/(2^(2/7)*7^(6/7)))*(((29*(-856512704956195*2^(2/3) - 856512704956195*I*2^(2/3)*Sqrt[3] - 236175961274505*I*2^(2/3)*Sqrt[7] + 236175961274505*2^(2/3)*Sqrt[21] - 82410846*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) - 22757210*I*Sqrt[7]*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) - 2^(1/3)*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3) + I*2^(1/3)*Sqrt[3]*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3)))/(3*(-28315078360483963721967 - 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 - 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3)))^(1/7)* E^((2/7)*I*Pi*(4 + 2*#1))) + (1/(2^(2/7)*7^(6/7)))* ((-((29*(856512704956195*2^(2/3) + 856512704956195*I*2^(2/3)*Sqrt[3] - 236175961274505*I*2^(2/3)*Sqrt[7] + 236175961274505*2^(2/3)*Sqrt[21] + 82410846*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) - 22757210*I*Sqrt[7]* (-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3) + 2^(1/3)*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3) - I*2^(1/3)*Sqrt[3]*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(2/3)))/(3*(-28315078360483963721967 + 14567661764710372864621*I*Sqrt[7] + 87*Sqrt[42*(207920806074244041696261258325895199249 + 94909358977115370199737368310473890157*I*Sqrt[7])])^(1/3))))^(1/7)* E^((2/7)*I*Pi*(3 + 3*#1))) & I have a slightly smaller expression for Cos[2Pi/29] - Cos[5Pi/29], etc. --rwg 2011/8/24 Bill Gosper <billgosper@gmail.com>
Really From: tpiezas@gmail.com
Hello all,
The octic,
x^8-x^7+29x^2+29 = 0 (eq.1)
(by Igor Schein) is solvable, but not as easy as merely factoring over a square root extension. Rather, this can be solvable by the 29th root of unity.
My solution to (eq.1) is, [tweaked (with permission) by rwg on the basis of numerical evidence: {8 x} = {1-a-b+c+d-e-f-g, 1+a+b+c-d-e-f+g, 1-a+b-c-d+e-f-g, 1+a-b-c+d+e-f+g, 1-a-b-c-d-e+f+g, 1+a+b-c+d-e+f-g, 1+a-b+c-d+e+f-g, 1-a+b+c+d+e+f+g} ] where each {a,b,c,d,e,f,g} = Sqrt[4v_i+1] and the v_i are the 7 roots of the septic,
8903+47647v+39672v^2+7192v^3-522v^4-174v^5+v^7 = 0 (eq.2)
The solution of which was given by Peter Montgomery as,
v_i = 2(w^11+w^13+w^16+w^18)-2(w+w^12+w^17+w^28)-(w^2+w^5+w^24+w^27)+ (w^3+w^7+w^22+w^26)+(w^4+w^10+w^19+w^25)-(w^8+w^9+w^20+w^21) and one can set w_i = {t, t^7, t^23, t^25, t^16, t^20, t^24}, and t = exp(2Pi*I/29). P.S. Similarly, a solvable 32-deg equation in x can be solved by a 31-deg Lagrange resolvent in z in the form,
x = z1^(1/2) +/- z2^(1/2) +/- ... +/- z31^(1/2)
though, unfortunately, no explicit examples are yet known.
- Tito Piezas III http://sites.google.com/site/tpiezas/ (Check this out! --rwg)
[rwg: the septic roots are all real:
v->-2 (2 cos((2 π)/29)+2 cos((3 π)/29)+cos((4 π)/29)-2 cos((5 π)/29)-cos((6 π)/29)+2 cos((7 π)/29)-sin(π/58)-sin((3 π)/58)-sin((7 π)/58)+sin((9 π)/58)+sin((11 π)/58)-sin((13 π)/58)) v->2 (cos(π/29)+cos((2 π)/29)-cos((4 π)/29)-cos((5 π)/29)-2 cos((6 π)/29)-2 sin(π/58)-sin((3 π)/58)-sin((5 π)/58)-sin((7 π)/58)-sin((9 π)/58)-2 sin((11 π)/58)+2 sin((13 π)/58)) v->2 (2 cos(π/29)-cos((2 π)/29)-cos((3 π)/29)+cos((4 π)/29)+cos((5 π)/29)-cos((7 π)/29)-2 sin((3 π)/58)-2 sin((5 π)/58)-2 sin((7 π)/58)+sin((9 π)/58)+sin((11 π)/58)-sin((13 π)/58)) v->-2 (2 cos(π/29)-cos((2 π)/29)+cos((3 π)/29)+cos((5 π)/29)+cos((6 π)/29)+cos((7 π)/29)+sin(π/58)-sin((3 π)/58)-2 sin((5 π)/58)-sin((7 π)/58)-2 sin((11 π)/58)+2 sin((13 π)/58)) v->-2 (cos(π/29)+cos((2 π)/29)-2 cos((3 π)/29)-2 cos((4 π)/29)-cos((5 π)/29)+cos((6 π)/29)-2 cos((7 π)/29)+sin(π/58)-sin((5 π)/58)-2 sin((9 π)/58)+sin((11 π)/58)-sin((13 π)/58)) v->2 (cos(π/29)+2 cos((2 π)/29)+cos((3 π)/29)+cos((4 π)/29)-2 cos((5 π)/29)+cos((6 π)/29)+cos((7 π)/29)+sin(π/58)+2 sin((3 π)/58)-sin((5 π)/58)+2 sin((7 π)/58)+sin((9 π)/58)) v->-2 (cos(π/29)-cos((3 π)/29)+2 cos((4 π)/29)-2 cos((6 π)/29)-cos((7 π)/29)-2 sin(π/58)+sin((3 π)/58)-sin((5 π)/58)+sin((7 π)/58)+2 sin((9 π)/58)-sin((11 π)/58)+sin((13 π)/58))]