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It is known that the equation $x^2+Dy^2=1$ is solved over finite fields $\mathbb{F}_q$ and we can point out the solutions . I wonder can we give solutions for the equation $x^2+Dy^2=\alpha$ for any $\alpha$ in $\mathbb{F}_q$?

Tricker
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  • You are interested in the solution of this equation in rational numbers? – individ Jun 17 '15 at 13:02
  • You'd have to find one "obvious" solution to find all of them. If $\alpha$ is a perfect square, and you can find its square root, you are done, but that is not trivial. – Thomas Andrews Jun 17 '15 at 13:05
  • ThomasAndrews: yes, because the solution of the first equation is the kernel of the norm map. But how can we point out a nontrivial solution for the second? – Tricker Jun 17 '15 at 13:09
  • I guess I realized that You need. You can use this formula. http://math.stackexchange.com/questions/738446/solutions-to-ax2-by2-cz2/738527#738527 – individ Jun 17 '15 at 13:45

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