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I'm trying to show whether the form $x_1^2 + x_2^2 -15(x_3^2+x_4^2)$ is isotropic over $\mathbb{Q}$. I tried to apply the strong Hasse principle, but I don't get how to show isotropicity over $\mathbb{Q}_p$

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    $\begingroup$ For $p\nmid 6$ it follows from Hensel lemma and that the reduction $\bmod p$ is isotropic. For $p=\infty$ it is obvious, so it remains to check $p=2,3$. Won't it be anisotropic at $p=3$? $\endgroup$
    – reuns
    Apr 6, 2022 at 21:33
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    $\begingroup$ if $ 3 | r^2 + s^2 $ in integers, then both $3|r$ and $3|s.$ Start with $r=x_1 and $s=x_2$ in your form $\endgroup$
    – Will Jagy
    Apr 6, 2022 at 21:52

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