Working in F₁₇ (integers mod 17) with ω = 9. The 8th roots of unity are the powers of ω: each one raised to the 8th power gives 1. Multiplying by ω steps to the next row, so P(ωx) vs P(x) compares adjacent trace rows.
F₁₇
ω = 9
ω
P(ωx)
P(x)