Puzzle: The Parity of Quadratic Residues on a Cubic Curve

Let ppp be an odd prime with p≡2(mod3)p \equiv 2 \pmod 3p≡2(mod3). Define the set

Sp:={x∈Fp:x3+1≠0}.S_p := \{x \in \mathbb{F}_p : x^3+1 \neq 0\}.Sp​:={x∈Fp​:x3+1=0}.

Let χ:Fp→{−1,0,1}\chi:\mathbb{F}_p\to\{-1,0,1\}χ:Fp​→{−1,0,1} be the Legendre symbol (quadratic character), with χ(0)=0\chi(0)=0χ(0)=0.

Define the quantity

T(p)  :=  ∑x∈Spχ(x3+1).T(p) \;:=\; \sum_{x\in S_p} \chi(x^3+1).T(p):=x∈Sp​∑​χ(x3+1).

Your tasks

1. Show that T(p)T(p)T(p) is an integer in the range [−(p−1), p−1][-(p-1),\, p-1][−(p−1),p−1] (easy warm-up)