The word "algebra" is a distortion of the Arabic title of a treatise by al-Khwārizmī about algebraic methods. In modern usage, algebra has several meanings.

One use of the word "algebra" is the abstract study of number systems and operations within them, including such advanced topics as groups, rings, invariant theory, and cohomology. This is the meaning mathematicians associate with the word "algebra." When there is the possibility of confusion, this field of mathematics is often referred to as abstract algebra.

The word "algebra" can also refer to the "school algebra" generally taught in American middle and high schools. This includes the solution of polynomial equations in one or more variables and basic properties of functions and graphs. Mathematicians call this subject "elementary algebra," "high school algebra," "junior high school algebra," or simply "school algebra," reserving the word "algebra" for the more advanced aspects of the subject.

Finally, the word is used in a third way, not as a subject area but as a particular type of algebraic structure. Formally, an algebra is a vector space V over a field F with a multiplication. The multiplication must be distributive and, for every f in F and x,y in V must satisfy


An algebra is sometimes implicitly assumed to be associative or to possess a multiplicative identity.

Examples of algebras include the algebra of real numbers, vectors and matrices, tensors, complex numbers, and quaternions. (Note that linear algebra, which is the study of linear sets of equations and their transformation properties, is not an algebra in the formal sense of the word.) Other more exotic algebras that have been investigated and found to be of interest are usually named after one or more of their investigators. This practice unfortunately leads to entirely unenlightening names which are commonly used by algebraists without further explanation or elaboration.

See also

Abstract Algebra, Alternative Algebra, Associative Algebra, Banach Algebra, Boolean Algebra, Borel Sigma-Algebra, C-*-Algebra, Cayley Algebra, Clifford Algebra, Commutative Algebra, Derivation Algebra, Exterior Algebra, Fundamental Theorem of Algebra, Graded Algebra, Hecke Algebra, Heyting Algebra, Homological Algebra, Hopf Algebra, Jordan Algebra, Lie Algebra, Linear Algebra, Measure Algebra, Nonassociative Algebra, Power Associative Algebra, Quaternion, Robbins Algebra, Schur Algebra, Semisimple Algebra, Sigma-Algebra, Simple Algebra, Steenrod Algebra, Umbral Algebra, von Neumann Algebra Explore this topic in the MathWorld classroom

Portions of this entry contributed by John Renze

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Renze, John and Weisstein, Eric W. "Algebra." From MathWorld--A Wolfram Web Resource.

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