# Local analysis

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{{Short description|Mathematical theories}}
In [algebraic geometry](/source/algebraic_geometry) and related areas of [mathematics](/source/mathematics), '''local analysis''' is the practice of looking at a problem relative to each [prime number](/source/prime_number) ''p'' first, and then later trying to integrate the information gained at each prime into a 'global' picture. These are forms of the [localization](/source/%3ACategory%3ALocalization_(mathematics)) approach.

==Group theory==
In [group theory](/source/group_theory), local analysis was started by the [Sylow theorems](/source/Sylow_theorems), which contain significant information about the structure of a [finite group](/source/finite_group) ''G'' for each prime number ''p'' dividing the order of ''G''. This area of study was enormously developed in the quest for the [classification of finite simple groups](/source/classification_of_finite_simple_groups), starting with the [Feit–Thompson theorem](/source/Feit%E2%80%93Thompson_theorem) that groups of odd order are [solvable](/source/solvable_group).<ref>{{cite journal|first=Richard |last=Solomon |title=A Brief History of the Classification of the Finite Simple Groups |journal=Bulletin of the American Mathematical Society |volume=38 |number=3 |pages=315-352 |year=2001 |url=https://www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00909-0/S0273-0979-01-00909-0.pdf}}</ref>

==Number theory==
{{main|Localization of a ring}}

In [number theory](/source/number_theory) one may study a [Diophantine equation](/source/Diophantine_equation), for example, modulo ''p'' for all primes ''p'', looking for constraints on solutions.<ref>{{cite book|first=Henri |last=Cohen |title=Number Theory: Volume I: Tools and Diophantine Equations |pages=4-5 |publisher=Springer |year=2007 |series=[Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics) |isbn=978-0-387-49922-2}}</ref> The next step is to look modulo prime powers, and then for solutions in the [''p''-adic field](/source/p-adic_number). This kind of local analysis provides conditions for solution that are ''necessary''. In cases where local analysis (plus the condition that there are real solutions) provides also ''sufficient'' conditions, one says that the ''[Hasse principle](/source/Hasse_principle)'' holds: this is the best possible situation. It does for [quadratic form](/source/quadratic_form)s, but certainly not in general (for example for [elliptic curve](/source/elliptic_curve)s). The point of view that one would like to understand what extra conditions are needed has been very influential, for example for [cubic form](/source/cubic_form)s.

Some form of local analysis underlies both the standard applications of the [Hardy–Littlewood circle method](/source/Hardy%E2%80%93Littlewood_circle_method) in [analytic number theory](/source/analytic_number_theory), and the use of [adele ring](/source/adele_ring)s, making this one of the unifying principles across number theory.

== See also ==
* [:Category:Localization (mathematics)](/source/%3ACategory%3ALocalization_(mathematics))
* [Localization of a category](/source/Localization_of_a_category)
* [Localization of a module](/source/Localization_of_a_module)
* [Localization of a ring](/source/Localization_of_a_ring)
* [Localization of a topological space](/source/Localization_of_a_topological_space)
* [Hasse principle](/source/Hasse_principle)

==References==
{{reflist}}

Category:Number theory
Category:Finite groups
Category:Localization (mathematics)

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