Entry 8

The previous entry ended with $\cos\big(\tfrac{2\pi}{29}\big)$. This is not by Ramanujan, but I got reminded about something elegant regarding $p=29$ which he would have appreciated. Does anyone know why the octic found by Igor Schein in 1999  $$x^8-x^7+29x^2+29 = 0$$ is solvable in radicals, specifically by the $29$th root of unity? Any other octic or higher with a similarly simple form?

Incidentally, the discriminant of $x^8-x^7+ax^2+a$ is $$F(a)=186624 a^3 - 3561092 a^2 + 29511140 a - 7^7$$ and the only integer solution to the elliptic curve $F(a) = y^2$ is $a = 29$.