Local-global principle

A strategy in mathematics that studies a global problem through its versions over completions or localizations. It is especially powerful when local solvability at every place implies global solvability.

Completion (algebra): A construction that enlarges a ring or field by adjoining limits of Cauchy sequences in a chosen topology. The p-adic and real completions turn a global equation into separate local problems.
Hasse–Minkowski theorem: A theorem stating that a quadratic form over a number field is isotropic if and only if it is isotropic at every completion. It is a fundamental setting where local solvability completely decides global solvability.
Hasse principle failure: A case in which a variety has points over every completion of a number field but no rational point over the field. Such examples directly refute the inference from everywhere-local solvability to global solvability.
Number field: A finite extension of the rational numbers. Number fields are standard global settings whose completions supply arithmetic local fields.
Localization (commutative algebra): A construction that makes selected elements invertible, producing a ring that focuses on chosen prime ideals. Localization provides local rings in which global algebraic questions can be tested near individual primes.
Brauer–Manin obstruction: An obstruction to rational points on a variety, derived from its Brauer group and adelic points. It explains many failures of the Hasse principle that local points alone cannot detect.
Selmer curve: The genus-one curve 3x³ + 4y³ + 5z³ = 0, which has points over every completion of the rationals but no rational point. It is a classical concrete counterexample to the Hasse principle.
p-adic number: An element of the completion of the rationals under the p-adic absolute value. Each prime gives a nonarchimedean local field in which equations can be tested.
Place (mathematics): An equivalence class of absolute values on a field, including its archimedean and nonarchimedean valuations. Local-global tests usually range over the field's places, including finite primes and infinite places.
Selmer group: A finite or finitely generated group defined by imposing local conditions on global Galois cohomology classes. Its local conditions bound global rational points and help expose failures of local-global predictions.

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