algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

and related areas of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, local analysis is the practice 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 group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

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

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

''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 In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...

, starting with the Feit–Thompson theorem that groups of odd order are solvable.

Number theory

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

one may study a

Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...

, 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 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 t ...

s, but certainly not in general (for example for elliptic curves). 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 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 Dir ...

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

References

{{reflist Number theory Finite groups Localization (mathematics)

Group theory

Number theory

See also

References