Hasse–Minkowski theorem

In a nutshell

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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Importance

The importance of the Hasse–Minkowski theorem lies in the novel paradigm it presented for answering arithmetical questions: in order to determine whether an equation of a certain type has a solution in rational numbers, it is sufficient to test whether it has solutions over complete fields of real and p-adic numbers, where one can apply analytic techniques such as Newton's method and its p-adic analogue Hensel's lemma. This is the first significant example of a local-global principle, one of the most fundamental techniques in arithmetic geometry.

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Hasse–Minkowski theorem

Importance

Application to the classification of quadratic forms

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