Hasse-Minkowski theorem
= Hasse-Minkowski theorem
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{wiki=Hasse–Minkowski_theorem}
= Hasse-Minkowski principle
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A nondegenerate <quadratic form> over a <number field> is isotropic if and only if it is isotropic over every completion. Over $\mathbb Q$, these are the real field and all <P-adic numbers>. It turns a rational existence problem into local tests; denominators can then be cleared for an integral homogeneous equation.