Algebra

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'''Algebra''' is one of the main branches of [[Mathematics|mathematics]], together with [[Geometry|geometry]] and [[Analysis|analysis]]. In the form most people meet first, it is the study of mathematical symbols and the rules for manipulating them: where [[Arithmetic|arithmetic]] computes with specific numbers, algebra writes letters such as ''x'' and ''y'' for unknown quantities and reasons with them directly — from 1 + ''x'' = 0, the rules give ''x'' = −1, whatever ''x'' may stand for.<ref name="mathworld">"Algebra." ''Wolfram MathWorld''. [https://mathworld.wolfram.com/Algebra.html mathworld.wolfram.com/Algebra.html]. Retrieved July 2026.</ref>

The subject did not stop at equation solving. Its history is one of steady generalization: the letters came to stand for things other than numbers — vectors, matrices, symmetries — and eventually the rules themselves, rather than any particular objects obeying them, became the thing studied, in the axiomatic theory of algebraic structures. At that abstract end the word takes on its second sense: "an algebra" as a ''mathematical object'', most commonly a [[Vector space|vector space]] equipped with a multiplication of vectors (see [[Algebra#Algebra as a structure|below]]).

== Etymology and history ==

The word ''algebra'' comes from the Arabic ''al-jabr'' ("the reunion of broken parts"), taken from the title of the treatise ''al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala'' (''The Compendious Book on Calculation by Completion and Balancing''), written by the Persian mathematician [[Al-Khwarizmi|al-Khwārizmī]] around 820 CE.<ref name="boyer">Carl B. Boyer and Uta C. Merzbach, ''A History of Mathematics'' (3rd ed., Wiley, 2011).</ref>

Algebraic problem-solving is far older than the name. Babylonian scribes solved quadratic problems on clay tablets nearly four thousand years ago, and [[Diophantus]] of Alexandria treated equations with symbolic abbreviations in his ''Arithmetica'' (3rd century CE).<ref name="boyer" /> Modern symbolic notation emerged in Europe with [[François Viète|Viète]] in the late 16th century and [[René Descartes|Descartes]] in ''La Géométrie'' (1637), which established conventions still in use, such as ''x'', ''y'', ''z'' for unknowns.<ref name="boyer" />

In the 19th century the subject turned abstract: [[Évariste Galois|Galois]]'s work on the solvability of equations{{Note|That is, which polynomial equations can be solved by a formula built from arithmetic operations and root extractions, the way the quadratic formula solves every degree-2 equation. Abel had already shown that no such general formula exists from degree 5 upward; Galois determined exactly which individual equations admit one.}} founded [[Group (mathematics)|]] theory, and [[George Boole|Boole]] showed that logic itself could be treated algebraically. The systematic, axiomatic style of modern abstract algebra took shape in the early 20th century, above all in the school of [[Emmy Noether]].{{Note|The lineage is concretely documented: the axiomatic approach grew out of [[Richard Dedekind]]'s and [[David Hilbert]]'s work on ideal theory, was reshaped in the 1920s by Noether's school at Göttingen, and was codified in van der Waerden's ''Moderne Algebra'' (1930), whose foreword credits lectures of Noether and [[Emil Artin]].}}<ref name="boyer" /><ref name="vdw">B. L. van der Waerden, ''Moderne Algebra'' (Springer, 1930) — the foreword credits lectures of E. Artin and E. Noether.</ref>

== Branches ==

; [[Elementary algebra]]
: Symbolic arithmetic and its application to solving equations — the algebra of school curricula, including polynomials, factoring, and equations such as ''ax''<sup>2</sup> + ''bx'' + ''c'' = 0, along with the first study of [[Function (mathematics)|]]s and their graphs.
; [[Linear algebra]]
: The study of [[Vector space|vector space]]s and [[Linear map|linear map]]s between them, worked concretely through [[Matrix (mathematics)|]] computations. It underpins systems of linear equations, geometry in ''n'' dimensions, and much of applied mathematics.
; [[Abstract algebra]]
: The axiomatic study of algebraic structures such as [[Group (mathematics)|]]s, [[Ring (mathematics)|]]s, and [[Field (mathematics)|]]s — sets equipped with operations satisfying specified laws.<ref name="df">David S. Dummit and Richard M. Foote, ''Abstract Algebra'' (3rd ed., Wiley, 2004).</ref>
; [[Universal algebra]]
: The study of properties common to all algebraic structures at once, treating "an algebra" in full generality as a set with a family of operations.
; [[Homological algebra]]
: The study of algebraic structures through sequences of maps between them (chain complexes and their homology). Originally abstracted from algebraic topology, it is now a toolkit used across algebra and geometry.
; [[Computer algebra]]
: Algorithms and software for exact symbolic computation — manipulating polynomials, solving equations, and simplifying expressions mechanically rather than numerically.

Algebraic methods also drive whole neighboring fields, including [[Algebraic geometry|algebraic geometry]], [[Algebraic number theory|algebraic number theory]], and [[Algebraic combinatorics|algebraic combinatorics]].

== Algebra as a structure ==

In the second sense of the word, "an algebra" is a particular kind of [[Algebraic structure|algebraic structure]] — not a synonym for it. Any set with operations satisfying axioms (a group, a ring, a lattice) is an algebraic structure, but only a specific one of them is called an algebra: an '''algebra over a field''' is a vector space equipped with a bilinear multiplication of vectors — that is, a product that distributes over addition and is compatible with scaling.<ref name="df" /> The definition generalizes directly to an '''algebra over a commutative ring'''. (Only in [[Universal algebra|universal algebra]] is "an algebra" used in the fully general sense, as a synonym for any algebraic structure.)

=== The vector-space family ===

These are algebras in the sense just defined, distinguished by which laws the multiplication obeys and what extra structure is present:

* '''[[Associative algebra|Associative algebras]]''' — the multiplication is associative. The square matrices of a fixed size form the standard example.
* '''[[Lie algebra|Lie algebras]]''' — the product is an antisymmetric bracket satisfying the Jacobi identity rather than associativity. Introduced through Sophus Lie's study of continuous symmetry, they are central to modern geometry and physics.<ref name="df" /> [[Poisson algebra|Poisson algebras]] and [[Jordan algebra|Jordan algebras]] are other important families with non-associative products.
* '''[[Banach algebra|Banach algebras]]''' and their relatives — in functional analysis, associative algebras over the real or complex numbers that also carry a complete norm compatible with the product. Adding an involution ("adjoint") operation leads to [[C*-algebra|C*-algebras]] and [[Von Neumann algebra|von Neumann algebras]], the [[Operator algebra|operator algebras]] underlying quantum theory.<ref name="rudin">Walter Rudin, ''Functional Analysis'' (2nd ed., McGraw-Hill, 1991).</ref>

=== Name-sharing structures ===

A second class of "algebras" generalizes logical connectives, sets, and lattices rather than vector multiplication — these are not algebras in the vector-space sense at all:

* '''[[Boolean algebra|Boolean algebras]]''' — structures with the operations ''and'', ''or'', and ''not'', abstracted from Boole's algebra of logic and fundamental to computer science. [[Heyting algebra|Heyting algebras]] are their intuitionistic-logic counterpart, dropping the law of excluded middle.
* '''[[Relational algebra]]''' — operations on finitary relations such as selection, projection, and join; the theoretical foundation of database query languages.<ref name="codd">E. F. Codd, "A Relational Model of Data for Large Shared Data Banks." ''Communications of the ACM'' 13(6), 1970.</ref>
* '''Algebras of sets''' — in measure theory, an [[Algebra over a set|algebra over a set]] is a collection of sets closed under finite unions and complementation; a [[Sigma-algebra|sigma-algebra]] strengthens this to countable unions, providing the setting for measure and probability.

Category theory generalizes the word once more: an [[F-algebra]] packages "a set with operations" into a single arrow, a formulation widely used in theoretical computer science.

== Other uses ==

Outside mathematics, ''Algebra'' is also the stage name of the American R&B singer [[Algebra (singer)|Algebra Blessett]], and the title of a well-known graduate textbook by [[Serge Lang]], ''[[Algebra (book)|Algebra]]''.

== See also ==

* [[Geometry]]
* [[Analysis]]
* [[Arithmetic]]
* [[Number theory]]
* [[Mathematics]]

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'''Algebra''' is one of the main branches of [[Mathematics|mathematics]], together with [[Geometry|geometry]] and [[Analysis|analysis]]. In the form most people meet first, it is the study of mathematical symbols and the rules for manipulating them: where [[Arithmetic|arithmetic]] computes with specific numbers, algebra writes letters such as ''x'' and ''y'' for unknown quantities and reasons with them directly — from 1 + ''x'' = 0, the rules give ''x'' = −1, whatever ''x'' may stand for.<ref name="mathworld">"Algebra." ''Wolfram MathWorld''. [https://mathworld.wolfram.com/Algebra.html mathworld.wolfram.com/Algebra.html]. Retrieved July 2026.</ref> The subject did not stop at equation solving. Its history is one of steady generalization: the letters came to stand for things other than numbers — vectors, matrices, symmetries — and eventually the rules themselves, rather than any particular objects obeying them, became the thing studied, in the axiomatic theory of algebraic structures. At that abstract end the word takes on its second sense: "an algebra" as a ''mathematical object'', most commonly a [[Vector space|vector space]] equipped with a multiplication of vectors (see [[Algebra#Algebra as a structure|below]]). == Etymology and history == The word ''algebra'' comes from the Arabic ''al-jabr'' ("the reunion of broken parts"), taken from the title of the treatise ''al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala'' (''The Compendious Book on Calculation by Completion and Balancing''), written by the Persian mathematician [[Al-Khwarizmi|al-Khwārizmī]] around 820 CE.<ref name="boyer">Carl B. Boyer and Uta C. Merzbach, ''A History of Mathematics'' (3rd ed., Wiley, 2011).</ref> Algebraic problem-solving is far older than the name. Babylonian scribes solved quadratic problems on clay tablets nearly four thousand years ago, and [[Diophantus]] of Alexandria treated equations with symbolic abbreviations in his ''Arithmetica'' (3rd century CE).<ref name="boyer" /> Modern symbolic notation emerged in Europe with [[François Viète|Viète]] in the late 16th century and [[René Descartes|Descartes]] in ''La Géométrie'' (1637), which established conventions still in use, such as ''x'', ''y'', ''z'' for unknowns.<ref name="boyer" /> In the 19th century the subject turned abstract: [[Évariste Galois|Galois]]'s work on the solvability of equations{{Note|That is, which polynomial equations can be solved by a formula built from arithmetic operations and root extractions, the way the quadratic formula solves every degree-2 equation. Abel had already shown that no such general formula exists from degree 5 upward; Galois determined exactly which individual equations admit one.}} founded [[Group (mathematics)|]] theory, and [[George Boole|Boole]] showed that logic itself could be treated algebraically. The systematic, axiomatic style of modern abstract algebra took shape in the early 20th century, above all in the school of [[Emmy Noether]].{{Note|The lineage is concretely documented: the axiomatic approach grew out of [[Richard Dedekind]]'s and [[David Hilbert]]'s work on ideal theory, was reshaped in the 1920s by Noether's school at Göttingen, and was codified in van der Waerden's ''Moderne Algebra'' (1930), whose foreword credits lectures of Noether and [[Emil Artin]].}}<ref name="boyer" /><ref name="vdw">B. L. van der Waerden, ''Moderne Algebra'' (Springer, 1930) — the foreword credits lectures of E. Artin and E. Noether.</ref> == Branches == ; [[Elementary algebra]] : Symbolic arithmetic and its application to solving equations — the algebra of school curricula, including polynomials, factoring, and equations such as ''ax''<sup>2</sup> + ''bx'' + ''c'' = 0, along with the first study of [[Function (mathematics)|]]s and their graphs. ; [[Linear algebra]] : The study of [[Vector space|vector space]]s and [[Linear map|linear map]]s between them, worked concretely through [[Matrix (mathematics)|]] computations. It underpins systems of linear equations, geometry in ''n'' dimensions, and much of applied mathematics. ; [[Abstract algebra]] : The axiomatic study of algebraic structures such as [[Group (mathematics)|]]s, [[Ring (mathematics)|]]s, and [[Field (mathematics)|]]s — sets equipped with operations satisfying specified laws.<ref name="df">David S. Dummit and Richard M. Foote, ''Abstract Algebra'' (3rd ed., Wiley, 2004).</ref> ; [[Universal algebra]] : The study of properties common to all algebraic structures at once, treating "an algebra" in full generality as a set with a family of operations. ; [[Homological algebra]] : The study of algebraic structures through sequences of maps between them (chain complexes and their homology). Originally abstracted from algebraic topology, it is now a toolkit used across algebra and geometry. ; [[Computer algebra]] : Algorithms and software for exact symbolic computation — manipulating polynomials, solving equations, and simplifying expressions mechanically rather than numerically. Algebraic methods also drive whole neighboring fields, including [[Algebraic geometry|algebraic geometry]], [[Algebraic number theory|algebraic number theory]], and [[Algebraic combinatorics|algebraic combinatorics]]. == Algebra as a structure == In the second sense of the word, "an algebra" is a particular kind of [[Algebraic structure|algebraic structure]] — not a synonym for it. Any set with operations satisfying axioms (a group, a ring, a lattice) is an algebraic structure, but only a specific one of them is called an algebra: an '''algebra over a field''' is a vector space equipped with a bilinear multiplication of vectors — that is, a product that distributes over addition and is compatible with scaling.<ref name="df" /> The definition generalizes directly to an '''algebra over a commutative ring'''. (Only in [[Universal algebra|universal algebra]] is "an algebra" used in the fully general sense, as a synonym for any algebraic structure.) === The vector-space family === These are algebras in the sense just defined, distinguished by which laws the multiplication obeys and what extra structure is present: * '''[[Associative algebra|Associative algebras]]''' — the multiplication is associative. The square matrices of a fixed size form the standard example. * '''[[Lie algebra|Lie algebras]]''' — the product is an antisymmetric bracket satisfying the Jacobi identity rather than associativity. Introduced through Sophus Lie's study of continuous symmetry, they are central to modern geometry and physics.<ref name="df" /> [[Poisson algebra|Poisson algebras]] and [[Jordan algebra|Jordan algebras]] are other important families with non-associative products. * '''[[Banach algebra|Banach algebras]]''' and their relatives — in functional analysis, associative algebras over the real or complex numbers that also carry a complete norm compatible with the product. Adding an involution ("adjoint") operation leads to [[C*-algebra|C*-algebras]] and [[Von Neumann algebra|von Neumann algebras]], the [[Operator algebra|operator algebras]] underlying quantum theory.<ref name="rudin">Walter Rudin, ''Functional Analysis'' (2nd ed., McGraw-Hill, 1991).</ref> === Name-sharing structures === A second class of "algebras" generalizes logical connectives, sets, and lattices rather than vector multiplication — these are not algebras in the vector-space sense at all: * '''[[Boolean algebra|Boolean algebras]]''' — structures with the operations ''and'', ''or'', and ''not'', abstracted from Boole's algebra of logic and fundamental to computer science. [[Heyting algebra|Heyting algebras]] are their intuitionistic-logic counterpart, dropping the law of excluded middle. * '''[[Relational algebra]]''' — operations on finitary relations such as selection, projection, and join; the theoretical foundation of database query languages.<ref name="codd">E. F. Codd, "A Relational Model of Data for Large Shared Data Banks." ''Communications of the ACM'' 13(6), 1970.</ref> * '''Algebras of sets''' — in measure theory, an [[Algebra over a set|algebra over a set]] is a collection of sets closed under finite unions and complementation; a [[Sigma-algebra|sigma-algebra]] strengthens this to countable unions, providing the setting for measure and probability. Category theory generalizes the word once more: an [[F-algebra]] packages "a set with operations" into a single arrow, a formulation widely used in theoretical computer science. == Other uses == Outside mathematics, ''Algebra'' is also the stage name of the American R&B singer [[Algebra (singer)|Algebra Blessett]], and the title of a well-known graduate textbook by [[Serge Lang]], ''[[Algebra (book)|Algebra]]''. == See also == * [[Geometry]] * [[Analysis]] * [[Arithmetic]] * [[Number theory]] * [[Mathematics]]