In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the term local analysis has at least two meanings - both derived from the idea of looking at a problem relative to each prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

 p first, and then later trying to integrate the information gained at each prime into a 'global' picture. These are forms of the localization approach.

Group theory

In group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, local analysis was started by the Sylow theorems, which contain significant information about the structure of a finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

 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

Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...

, starting with the Feit–Thompson theorem

Feit–Thompson theorem

In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...

 that groups of odd order are solvable

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...

.

Number theory

In number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

 one may study a Diophantine equation

Diophantine equation

In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

, 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

P-adic number

In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

. 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

Hasse principle

In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number...

holds: this is the best possible situation. It does for quadratic form

Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....

s, but certainly not in general (for example for elliptic curve

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

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

Hardy-Littlewood circle method

In mathematics, the Hardy–Littlewood circle method is one of the most frequently used techniques of analytic number theory. It is named for G. H. Hardy and J. E...

 in analytic number theory

Analytic number theory

In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

, and the use of adele ring

Adele ring

In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....

s, making this one of the unifying principles across number theory.