How to solve the equation $x^6-2\varphi^5x^5+2\varphi x+\varphi^6=0$ in radicals using Mathematica?
where $\varphi$ is the golden ratio.
Solution is Ramanujan's Class Invariant $G_{125}$, Class Invariant is always algebraic, therefore this equation is solvable in radicals.
I try to use "Solve and Reduce ", but there is no "Radical" output
t = (1 + Sqrt[5])/2
Solve[x^6 - 2 t^5 x^5 + 2 t x + t^6 == 0, x, Reals]
out
{{x -> 1}, {x -> Root[1 - 20 #1 - 45 #1^2 - 70 #1^3 - 95 #1^4 - 118 #1^5 - 95 #1^6 - 70 #1^7 - 45 #1^8 - 20 #1^9 + #1^10 &, 2]}}