Algebra
This article is about the branch of mathematics. For other uses, see Algebra (disambiguation).
Algebra comes from Arabic word (al-jabr , الجبر), it means a branch of mathematics concerning the study of structure, relation, and quantity. 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.
Classification
Algebra may be divided roughly into the following categories:
Elementary algebra
in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra), also called second year and third year algebra
Abstract algebra
sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated; this includes, among other fields,
Linear algebra
in which the specific properties of vector spaces are studied (including matrices);
Universal algebra
in which properties common to all algebraic structures are studied.
Algebraic number theory
in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
Algebraic geometry
in its algebraic aspect.
Algebraic combinatorics
in which abstract algebraic methods are used to study combinatorial questions.
In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric structure (a metric or a topology) which is compatible with the algebraic structure. The list includes a number of areas of functional analysis:
Normed linear spaces
Banach spaces
Hilbert spaces
Banach algebras
Normed algebras
Topological algebras
Topological groups
This article is about the branch of mathematics. For other uses, see Algebra (disambiguation).
Algebra comes from Arabic word (al-jabr , الجبر), it means a branch of mathematics concerning the study of structure, relation, and quantity. 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.
Classification
Algebra may be divided roughly into the following categories:
Elementary algebra
in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra), also called second year and third year algebra
Abstract algebra
sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated; this includes, among other fields,
Linear algebra
in which the specific properties of vector spaces are studied (including matrices);
Universal algebra
in which properties common to all algebraic structures are studied.
Algebraic number theory
in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
Algebraic geometry
in its algebraic aspect.
Algebraic combinatorics
in which abstract algebraic methods are used to study combinatorial questions.
In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric structure (a metric or a topology) which is compatible with the algebraic structure. The list includes a number of areas of functional analysis:
Normed linear spaces
Banach spaces
Hilbert spaces
Banach algebras
Normed algebras
Topological algebras
Topological groups
