Local analysis

In Mathematical, the term analyzes local initially has at least two directions - both derived from the idea of examination of a problem relative to each Prime number p and, to later try to integrate the information gained on each prime number in a “total” table.

Theory of the groups

In Theory of the groups, the local analysis begin with the theorems from Sylow, which contain significant information on the structure of a Groupe finished G for each prime number p dividing the order of G . This field of study was enormously developed in the search of the classification of the finished simple groups, starting with the Théorème of Feit-Thompson these groups of an odd nature are resolvable.

Theory of the numbers

In Theory of the numbers, one can study a equation diophantienne, for example, modulo p for all the prime numbers p , seeking constraints on the solutions. The following stage is the examination modulo of the powers first, and finally, of the solutions in the body '' p '' - adic. This kind of local analysis provides conditions which are necessary for the solutions. Whenever the local analysis (more the condition which there exist real solutions) also provides of the sufficient conditions , it is said that the principle of Hasse is valid: it is the best possible situation. It is the case for the quadratic forms, but certainly not in general (for example for the elliptic curved ). The point of view which one would like to include/understand which additional conditions would be necessary was very influential, for example for the cubic forms.

Certain forms of local analysis are at the base of the standards applications of the Méthode of the circle of Hardy-Littlewood in the analytical Théorie of the numbers, and the use of the rings adèles, doing this one one of the unifying principles through the theory of the numbers.

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