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.

Reference

Serre, Jean Pierre (1973). A Course in Arithmetic. New York: Springer Verlag. ISBN 0-387-90040-3