- Local analysis
In

mathematics , the term**local analysis**has at least two meanings - both derived from the idea of looking at a problem relative to eachprime number "p" first, and then later trying to integrate the information gained at each prime into a 'global' picture.**Group theory**In

group theory , local analysis was started by theSylow theorems , which contain significant information about the structure of afinite group "G" for each prime number "p" dividing the order of "G". This area of study was enormously developed in the quest for theclassification of finite simple groups , starting with theFeit-Thompson theorem that groups of odd order are solvable...**Number theory**In

number theory one may study aDiophantine equation , for example, modulo "p" for all primes "p", looking for constraints on solutions. The next step is to look modulo prime powers, and then for solutions in the "p"-adic field. 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 " holds: this is the best possible situation. It does forquadratic form s, but certainly not in general (for example forelliptic 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 forms.Some form of local analysis underlies both the standard applications of the

Hardy-Littlewood circle method inanalytic number theory , and the use ofadele ring s, making this one of the unifying principles across number theory.

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