A nondegenerate quadratic form over a number field is isotropic if and only if it is isotropic over every completion. Over , these are the real field and all P-adic numbers. It turns a rational existence problem into local tests; denominators can then be cleared for an integral homogeneous equation.
For a nondegenerate diagonal quadratic form , this product of quadratic Hilbert symbols is independent of diagonalization. Over a p-adic field, dimension, determinant square class and this invariant classify the form up to isometry. The product of the local invariants of a globally diagonalized form is one by the Hilbert reciprocity law; real places additionally record signature.
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The Hasse–Minkowski theorem is a result in the field of number theory, specifically concerning the theory of quadratic forms. It establishes a fundamental connection between the local and global solvability of quadratic forms over the rational numbers. In simple terms, the theorem states that a quadratic form over the rational numbers can be represented by integers if and only if it can be represented by integers when considered over the completions of the rational numbers at all finite places and at infinity (the real numbers).