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(Redirected from 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 Hermann Minkowski and generalized to global fields by Helmut Hasse.
★ Serre, Jean Pierre (1973). ''A Course in Arithmetic''. New York: Springer Verlag. ISBN 0-387-90040-3.
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 Hermann Minkowski and generalized to global fields by Helmut Hasse.
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References
★ Serre, Jean Pierre (1973). ''A Course in Arithmetic''. New York: Springer Verlag. ISBN 0-387-90040-3.
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