Algebra

Algebra is one of the main branches of mathematics, together with geometry and analysis. In the form most people meet first, it is the study of mathematical symbols and the rules for manipulating them: where 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.^[1]

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 equipped with a multiplication of vectors (see below).

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-Khwārizmī around 820 CE.^[2]

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).^[2] Modern symbolic notation emerged in Europe with Viète in the late 16th century and Descartes in La Géométrie (1637), which established conventions still in use, such as x, y, z for unknowns.^[2]

In the 19th century the subject turned abstract: Galois's work on the solvability of equations^{[note 1]} founded Group theory, and 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 2][2][3]}

Elementary algebra

Symbolic arithmetic and its application to solving equations — the algebra of school curricula, including polynomials, factoring, and equations such as ax² + bx + c = 0, along with the first study of Functions and their graphs.

Linear algebra

The study of vector spaces and linear maps between them, worked concretely through Matrix 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 Groups, Rings, and Fields — sets equipped with operations satisfying specified laws.^[4]

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 number theory, and algebraic combinatorics.

In the second sense of the word, "an algebra" is a particular kind of 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.^[4] The definition generalizes directly to an algebra over a commutative ring. (Only in universal algebra is "an algebra" used in the fully general sense, as a synonym for any algebraic structure.)

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

• Associative algebras — the multiplication is associative. The square matrices of a fixed size form the standard example.
• 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.^[4] Poisson algebras and Jordan algebras are other important families with non-associative products.
• 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*-algebras and von Neumann algebras, the operator algebras underlying quantum theory.^[5]

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 algebras — structures with the operations and, or, and not, abstracted from Boole's algebra of logic and fundamental to computer science. 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.^[6]
• Algebras of sets — in measure theory, an algebra over a set is a collection of sets closed under finite unions and complementation; a 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.

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

• Geometry
• Analysis
• Arithmetic
• Number theory
• Mathematics

1. "Algebra." Wolfram MathWorld. mathworld.wolfram.com/Algebra.html. Retrieved July 2026.
2. Carl B. Boyer and Uta C. Merzbach, A History of Mathematics (3rd ed., Wiley, 2011).
3. B. L. van der Waerden, Moderne Algebra (Springer, 1930) — the foreword credits lectures of E. Artin and E. Noether.
4. David S. Dummit and Richard M. Foote, Abstract Algebra (3rd ed., Wiley, 2004).
5. Walter Rudin, Functional Analysis (2nd ed., McGraw-Hill, 1991).
6. E. F. Codd, "A Relational Model of Data for Large Shared Data Banks." Communications of the ACM 13(6), 1970.