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How to determine if quadratic form $q(x,y,z)=-6x^2+ \frac{7}{2}y^2-\frac{25}{7}z^2$ is anisotropic over $\mathbb{Q}$? This quadratic form is a diagonalisation of another quadratic form. Is there any criteria or a theorem that could give me the answer to the question? Any help is welcome.

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Just check that it's anisotropic over some $\Bbb Q_p$ (it is clearly isotropic over $\Bbb R$). The only $p$ that could possibly work are $p\in\{2,3,5,7\}$. (The Hasse-Minkowski theorem proves that if $q$ is isotropic locally then it is isotropic over $\Bbb Q$.)

Incidentally, $q$ is equivalent to $-6x^2+14y^2-7z^2$ over $\Bbb Q$. I think that's anisotropic over $\Bbb Q_3$.

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