mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

''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

group theory In 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 ...

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

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...

''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, starting with the

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. suggested using t ...

that groups of odd order are solvable.

Number theory

{{main, Localization of a ring In

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

one may study a

Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...

, 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 In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an diophantine equation, integer solution to an equation by using the Chinese remainder theorem to piece together solutions mod ...

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

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...

s, but certainly not in general (for example for

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...

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 In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic plane curve. In , Boris Delone and Dmitry ...

s. Some form of local analysis underlies both the standard applications of the Hardy–Littlewood circle method in

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 Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...

, and the use of adele rings, making this one of the unifying principles across number theory.

Group theory

Number theory

See also