Hasse–Minkowski theorem

A nondegenerate quadratic form over a number field has a nontrivial zero if and only if it has one over every completion of that field.

Quadratic form: A homogeneous polynomial of degree two in several variables. The theorem applies to these forms over number fields.
Nondegenerate quadratic form: A quadratic form whose associated symmetric bilinear form has zero radical. The theorem assumes nondegeneracy to apply the standard classification.
Helmut Hasse: A German mathematician who developed major parts of class field theory and arithmetic algebra. His work established the theorem's modern number-field formulation.
Representation of integers by quadratic forms: The study of which integers are values of a given quadratic form on integer vectors. Local tests can constrain whether an integer is represented globally.
Failure of the Hasse principle: A global equation has solutions over every completion of a field but no global solution. Such failures occur for broader classes of equations, unlike quadratic forms here.
Number field: A finite extension of the rational numbers. It supplies the global field whose completions are tested.
Isotropy of a quadratic form: The property that a quadratic form vanishes on a nonzero vector. This is the precise meaning of having a nontrivial zero.
Minkowski–Hasse theorem for quadratic forms: A name sometimes used for the local-global theorem on quadratic forms over number fields. The alternative naming credits Minkowski alongside Hasse.
Conic over a number field: A smooth plane curve defined by a homogeneous quadratic equation over a number field. A conic has a rational point exactly when its associated ternary form is isotropic.
Mordell curve: An elliptic curve given by an equation of the form y² = x³ + k. Some curves of this kind exhibit failures of local-global solvability.

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