Hasse-Minkowski theorem

In mathematics, the Hasse-Minkowski theorem states that a quadratic form is isotropic globally if and only if it is everywhere isotropic locally; it is the classic local-global principle. Here to be isotropic means to that there is some non-zero vector for which the quadratic form returns zero as a value. Isotropic globally means there is a global field, ie either a number field or a function field over a finite field, over which the quadratic form is defined and is isotropic. Isotropic locally means that for every completion, both archimedean and non-archimedean, the quadratic form is isotropic.

The theorem was proven in the special case of the rational numbers by Minkowski and generalized to global fields by Hasse.

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