Number of solutions in a field of order $32$

Question

Let $F$ be a field of order $32$. Then find the number of non-zero solutions $(a,b)\in F\times F$ of the equation $x^2+xy+y^2=0$.

As , $|F|=32$ , so $(F\setminus\{0\},.)$ forms a group of order $31$, which is prime. So , $F\setminus \{0\}\simeq\mathbb Z_{31}$.

Then how I proceed ?

Answer 1

If $x^2 + xy + y^2=0$, then $x^3 - y^3 = (x-y)(x^2+xy+y^2)=0$.

But $3$ does not divide $31$, so $a\mapsto a^3$ is injective, and therefore $x=y$.