Foundations, part 9

$\begin{aligned} \chi & = z^{86} + z^{175} + z^{289} + z^{918} + z^{2017} \\ & = z^{{\color{#3D99F6}14} \times 6+2} + z^{{\color{#3D99F6}14} \times12 +7} + z^{{\color{#3D99F6}14} \times 20+9} + z^{{\color{#3D99F6}14} \times 65+8} + z^{{\color{#3D99F6}14} \times 144+1} & \small \color{#3D99F6} \text{Note that }z = \sqrt[7]{-1} \implies z^7 = -1 \implies z^{14} = 1 \\ & = z^2 + z^7 + z^9 + z^8 + z^1 \\ & = z^2 + z^{\color{#3D99F6}7} + z^{{\color{#3D99F6}7}+2} + z^{{\color{#3D99F6}7}+1} + z^1 & \small \color{#3D99F6} \text{Note that }z^7 = -1 \\ & = z^2 + {\color{#3D99F6}-1} {\color{#3D99F6}-}z^2 {\color{#3D99F6}-}z^1 + z^1 \\ & = \boxed{-1} \end{aligned}$

Since $\large z$ is a non-real root of $\large \sqrt[7]{-1}$ it implies $\large z^7 = -1$

Now $\large z^{86} = (z^7)^{12}\times z^2 = (-1)^{12}\times z^2 = z^2$

$\large z^{175} = (z^7)^{25} = (-1)^{25} = -1$

$\large z^{289} = (z^7)^{41}\times z^2 = (-1)^{41}\times z^2 = -z^2$

$\large z^{918} = (z^7)^{131}\times z = (-1)^{131}\times z = -z$

$\large z^{2017} = (z^7)^{288}\times z = (-1)^{288}\times z = z$

Adding all of these will get $\large z^2 + (-1) + (-z^2) + (-z) + z = \boxed {-1}$