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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 muslim Persian mathematician, astronomer, astrologer and geographer, Muhammad bin Mūsā al-Khwārizmī titled Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations.

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.

Algebra is much broader than elementary algebra and can be generalized. 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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Example of a Persian design with wallpaper group type "p6m"

A wallpaper group is a mathematical concept used to classify repetitive designs on two-dimensional surfaces, such as floors and walls, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. The mathematical study of such patterns reveals that exactly 17 different types of pattern can occur.

Wallpaper groups are examples of an abstract algebraic structure known as a group. Groups are frequently used in mathematics to study the notion of symmetry. Wallpaper groups are related to the simpler frieze groups, and to the more complex three-dimensional crystallographic groups.

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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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Pictures of all the connected Dynkin diagrams

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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Topics in algebra

General Elementary algebra Key concepts Linear algebra
  • Elementary algebra
  • Function (mathematics)
  • Equation
  • Square root, cube root, Nth root
  • Polynomial
    • Linear equation
    • Quadratic equation
    • Cubic equation
  • Simultaneous equations
  • Partial fraction
  • Associativity, Commutativity
  • 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
  • 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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