Hasse–Minkowski theorem
The importance of the Hasse–Minkowski theorem lies in the novel paradigm it presented for answering arithmetical questions: in order to determine whether an equation of a certain type has a solution in rational numbers, it is sufficient to test whether it has solutions over complete fields of real and p-adic numbers, where one can apply analytic techniques such as Newton's method and its p-adic analogue Hensel's lemma.This is the first significant example of a local-global principle, one of the most fundamental techniques in arithmetic geometry.The Hasse–Minkowski theorem reduces the problem of classifying quadratic forms over a number field K up to equivalence to the set of analogous but much simpler questions over local fields.Basic invariants of a nonsingular quadratic form are its dimension, which is a positive integer, and its discriminant modulo the squares in K, which is an element of the multiplicative group K*/K*2.Depending on the choice of v, this completion may be the real numbers R, the complex numbers C, or a p-adic number field, each of which has different kinds of invariants: These invariants must satisfy some compatibility conditions: a parity relation (the sign of the discriminant must match the negative index of inertia) and a product formula (a local–global relation).