on peut trouver des solutions au degre 5
soit a0 = -(3*(113*sin(arctan(1033*(367)^(1/2)/11744)/3+pi/6)*3^(1/2)-15*(113)^(1/2))*cos(arctan(1033*(367)^(1/2)/11744)/3+pi/6)-339*(sin(arctan(1033*(367)^(1/2)/11744)/3+pi/6))^2+45*sin(arctan(1033*(367)^(1/2)/11744)/3+pi/6)*(339)^(1/2)-421)/800
a1 = (-3*cos(arctan(1033*(367)^(1/2)/11744)/3+pi/6)*(113)^(1/2))/80+3*sin(arctan(1033*(367)^(1/2)/11744)/3+pi/6)*(339)^(1/2)/80-11/80
a2 = (3*cos(arctan(1033*(367)^(1/2)/11744)/3+pi/6)*(113)^(1/2)-3*sin(arctan(1033*(367)^(1/2)/11744)/3+pi/6)*(339)^(1/2)+35)/40
alors l'équation x^5+x^4+x^3+a2*x^2+a1*x+a0=0 admet comme solution :
x = 1/10*sqrt(5*3^(2/5)*(63845*sin(1/6*pi + 1/3*arctan(1033/11744*sqrt(367)))^4 - 24860*sqrt(339)*sin(1/6*pi + 1/3*arctan(1033/11744*sqrt(367)))^3 + 565*(339*sin(1/6*pi + 1/3*arctan(1033/11744*sqrt(367)))^2 - 44*sqrt(339)*sin(1/6*pi + 1/3*arctan(1033/11744*sqrt(367))) + 619)*cos(1/6*pi + 1/3*arctan(1033/11744*sqrt(367)))^2 - 10*(12769*sqrt(3)*sin(1/6*pi + 1/3*arctan(1033/11744*sqrt(367)))^3 - 9944*sqrt(113)*sin(1/6*pi + 1/3*arctan(1033/11744*sqrt(367)))^2 + 92321*sqrt(3)*sin(1/6*pi + 1/3*arctan(1033/11744*sqrt(367))) - 5139*sqrt(113))*cos(1/6*pi + 1/3*arctan(1033/11744*sqrt(367))) + 1272945*sin(1/6*pi + 1/3*arctan(1033/11744*sqrt(367)))^2 - 51390*sqrt(339)*sin(1/6*pi + 1/3*arctan(1033/11744*sqrt(367))) + 218727)^(1/5) - 15) - 1/2
on le trouve avec une racine ^(1/5)