Portal:Algebra

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Algebra

Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Persian mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr al-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided operations for the systematic solution of linear and quadratic equation

Together with geometry, analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics. Elementary algebra is often part of the curriculum in secondary education and provides an introduction to the basic ideas of algebra, including effects of adding and multiplying numbers, the concept of variables, definition of polynomials, along with factorization and determining their roots.

In addition to working directly with numbers, algebra covers working with symbols, variables, and set elements. Addition and multiplication are viewed as general operations, and their precise definitions lead to structures such as groups, rings and fields.

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A Hilbert space is a real or complex vector space with a positive-definite Hermitian form, that is complete under its norm. Thus it is an inner product space, which means that it has notions of distance and of angle (especially the notion of orthogonality or perpendicularity). The completeness requirement ensures that for infinite dimensional Hilbert spaces the limits exist when expected, which facilitates various definitions from calculus. A typical example of a Hilbert space is the space of square summable sequences.

Hilbert spaces allow simple geometric concepts, like projection and change of basis to be applied to infinite dimensional spaces, such as function spaces. They provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics.

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Abstract algebra • Algebraic geometry • Algebraic topology • Commutative algebra • Computer algebra • Diagram algebras • Elementary algebra • Galois theory • Homological algebra • Linear algebra • Mathematical identities • Permutations • Polynomials • Symmetric functions • Algebra stubs

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These are all the connected Dynkin diagrams, which classify the irreducible root systems, which themselves classify simple complex Lie algebras and simple complex Lie groups. These diagrams are therefore fundamental throughout Lie group theory.

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Did you know?

• ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
• ...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.
• ...that the Gudermannian function relates the regular trigonometric functions and the hyperbolic trigonometric functions without the use of complex numbers?
• ...that the classification of finite simple groups was not completed until the mid 1980s?
• ...that a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers?

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Topics in algebra

General	Elementary algebra	Key concepts	Linear algebra
Algebra Abstract algebra Algebraic number theory Algebraic geometry Topological algebra Algebraic combinatorics Commutative algebra Homological algebra Abstract algebra topics	Elementary algebra Function (mathematics) Equation Exponent Square root, cube root, Nth root Logarithms Polynomial Linear equation Quadratic equation Cubic equation Simultaneous equations Partial fraction	Identity element Inverse element Commutativity Associativity Distributivity Isomorphism The Isomorphism theorems Homomorphism	Linear algebra Matrix, Matrix theory Determinant Vector, Vector space Inner product, Inner product space Hilbert space Linear algebra glossary Linear algebra topics
Algebraic structures	Groups	Rings and Fields	Other
Algebraic structure (list) Group, Group theory Magma, quasigroup, loop, semigroup, monoid Lattice Rng, Semiring Ring, Ring theory Field, Field theory Module, Comodule Algebra over a ring, Algebra over a field Lie algebra Jordan algebra Universal algebra	Group, Group theory Order Subgroup, Normal subgroup Abelian group Simple group Finite simple groups Classification of finite simple groups Lie group Symmetry group Group representation Computational group theory Topological group Glossary of group theory List of group theory topics	Ring, Ring theory Field, Field theory Factor ring Ideal Ring homomorphism Banach algebra Clifford algebra Glossary of ring theory Glossary of field theory	Order theory Well-ordered Domain theory Glossary of order theory List of order topics Category theory (Portal) Tensor algebra K-theory Module theory Representation theory Computer algebra system Normed algebra

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Algebra	Analysis	Category theory	Computer science	Cryptography	Discrete mathematics	Geometry

Logic	Mathematics	Number theory	Physics	Science	Set theory	Statistics	Topology

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