Can you show that one form of the solution is the following: Sec 2π/17 = (2+√17+√W-√Y) /2

Question

It is known that Sec2π/17 radians is a root of the equation: x8 -8x7 - 40X6 + 80X5 + 240X4 -192X3 - 448X2 +128x + 256 = 0 where X= Sec 2π/17. This is associated with the regular 17-sided polygon. This equation can be solved algebraically. Can you demonstrate this? Can you show that one form of the solution is the following: Sec 2π/17 = (2+√17+√W-√Y) /2 , where W=17+4√17 , Y=2(17+2√17+4√W-√Z) , and Z=17-4√17.

Answer 1

Well, the nicer form of this comes from taking $2 \cos \frac{2 \pi}{17}$ in the form $\omega + \omega^{-1},$ where $\omega$ is a primitive 17th root of unity; taking the long factor out of $ \frac{x^{17} -1}{x-1}$ as zero leads to $$ x^8 + x^7 - 7 x^6 - 6 x^5 + 15 x^4 + 10 x^3 - 10 x^2 - 4x + 1 . $$ Writing it using only square roots is still a mess. It is done in Galois Theory by David A. Cox.