Number of elements in $ \{(x,y)\in \Bbb F_q^2:y^p+y=x^{p+1} \} $ where $p>2$ is a prime and $q=p^2$

Question

Let $p$ be a prime number $>2$ and $q=p^2$. Let $\Bbb F_q$ be a field of $q$ elements. Then it is easily shown that $\Bbb F_q=\{a+b\alpha : a,b\in \Bbb F_p, \alpha^2=u\}$, where $u$ is an element of $\Bbb F_p$ satisfying $u^{(p-1)/2}=-1$. I am trying to find the number of the elements of the set $$ \{(x,y)\in \Bbb F_q^2:y^p+y=x^{p+1} \} .$$ By the description of $\Bbb F_q$ above, we have $(a+b\alpha)^p+(a+b\alpha)=2a$ and $(c+d\alpha)^{p+1}=c^2-d^2u$. Thus the problem reduces to find the number of elements $a,b,c,d\in \Bbb F_q$ such that $2a=c^2-d^2u$ but I got stuck here. Any hints?

Answer 1

You can take $b,c,d$ to be arbitrary in $\mathbb{F}_{p}$, and then $a$ is fixed.