Complex number proof of solvability of regular 17 sided ruler and gauge drawing

The problem is equivalent to x^17=1, It can be solved by root equation .

First, let's look at the regular pentagon ,x^5=1

(x-1)(1+x+x^2+x^3+x^4)=0

x + x^2 + x^3 + x^4 = -1

among x = exp(2*pi/5)

Calculation （x + x^4) (x^2 + x^3) = x^3 + x^4 + x^6 + x^7 = x + x^2 + x^3 + x^4 = -1

The basis of grouping is that conjugate complex numbers must be put together .

Regular seventeen sided

x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 

x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 = -1

How to group for the first time ？

Conjugate complex numbers must be together . Shared after Xiangcheng 64 term , The same result will be achieved after simplification .

Guess multiply and equal -4.