Number of solutions for the given equation in finite field of order 32.

Question

Let $F$ be a field of order 32. Find number of solutions $(x,y)\in F\times F$ for $x^2 +y^2 +xy =0$.

I have figured out that non zero elements of this field forms a cyclic multiplicative group of order 31 and charecteristic of $ F $ must be 2. Have no idea how to proceed further.

Answer 1

If $y = 0$, clearly also $x = 0$.

So suppose $y \ne 0$, multiply by $y^{-2}$, and set $t = x y^{-1}$ to obtain $t^{2} + t + 1 = 0$.

Now you should know that the solutions of the latter equations are the two elements different from $0, 1$ in the field of order $2^{2} = 4$. Since $32 = 2^{5}$, the field of order $4$ is not a subfield of $F$. So no solutions here.

Answer 2

Hint $$x^3-y^3=(x-y)(x^2+xy+y^{2})$$

Hint 2

If $x \neq 0$ and $y \neq 0$ you get $$x^{3}=y^3 \\ x^{31}=y^{31}$$

From here you get immediately $x=y$, which you can plug in the original equation.

The case $x=0$ OR $y=0$ is easy...