Let $F$ be a simple transcendental extension of the field with three elements. How can I find $a, b \in F$ such that the quadratic form $x^2-ay^2-bz^2$ is anisotropic?

  • $\begingroup$ Not very much at this stage. Should I just try a few examples and try to work out whether the quadratic form is anisotropic or not and see if I can get some intuition for the problem that way? $\endgroup$ – Rupert Sep 7 '16 at 4:31
  • $\begingroup$ Probably not a bad idea. Also this might not help, but maybe if you phrase the question as finding out for which $a,b$ does $x^2−ay^2−bz^2$ have roots over $\mathbb{F}_3(\alpha)$, you might be able to entice an arithmetic geometry type to answer? $\endgroup$ – A. S. Sep 7 '16 at 5:09

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