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?
• 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? – A. S. Sep 7 '16 at 5:09