Hasse–Minkowski theorem

Two completions of the rational numbers, the dyadic numbers (here, only the dyadic integers are shown) and the real numbers. The Hasse-Minkowski theorem gives a relationship between quadratic forms in a number field and in the completions of the number field.

The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every topological completion of the field (which may be real, complex, or p-adic). A related result is that a quadratic space over a number field is isotropic if and only if it is isotropic locally everywhere, or equivalently, that a quadratic form over a number field nontrivially represents zero if and only if this holds for all completions of the field. The theorem was proved in the case of the field of rational numbers by Hermann Minkowski and generalized to number fields by Helmut Hasse. The same statement holds even more generally for all global fields.

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