 TOPICS  # Algebra

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 over a field with a multiplication. The multiplication must be distributive and, for every and 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.

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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## References

Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991.Bhattacharya, P. B.; Jain, S. K.; and Nagpu, S. R. (Eds.). Basic Algebra, 2nd ed. New York: Cambridge University Press, 1994.Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, 1996.Cardano, G. Ars Magna or The Rules of Algebra. New York: Dover, 1993.Chevalley, C. C. Introduction to the Theory of Algebraic Functions of One Variable. Providence, RI: Amer. Math. Soc., 1951.Chrystal, G. Textbook of Algebra, 2 vols. New York: Dover, 1961.Dickson, L. E. Algebras and Their Arithmetics. Chicago, IL: University of Chicago Press, 1923.Dickson, L. E. Modern Algebraic Theories. Chicago: H. Sanborn, 1926.Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998.Edwards, H. M. Galois Theory, corrected 2nd printing. New York: Springer-Verlag, 1993.Euler, L. Elements of Algebra. St. Petersburg, Russia: Royal Acad. Sci., 1770. English reprint Sangwin, C. (Ed.). Stradbroke, England: Tarquin Pub., 2007.Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lexington, MA: D. C. Heath, 1994.Grove, L. Algebra. New York: Academic Press, 1983.Hall, H. S. and Knight, S. R. Higher Algebra, A Sequel to Elementary Algebra for Schools. London: Macmillan, 1960.Harrison, M. A. "The Number of Isomorphism Types of Finite Algebras." Proc. Amer. Math. Soc. 17, 735-737, 1966.Herstein, I. N. Noncommutative Rings. Washington, DC: Math. Assoc. Amer., 1996.Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 1975.Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. Freeman, 1989.Kaplansky, I. Fields and Rings, 2nd ed. Chicago, IL: University of Chicago Press, 1995.Lang, S. Undergraduate Algebra, 2nd ed. New York: Springer-Verlag, 1990.MathPages. "Algebra." http://www.mathpages.com/home/ialgebra.htm.Pedersen, J. "Catalogue of Algebraic Systems." http://www.math.usf.edu/~eclark/algctlg/.Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, 1996.Spiegel, M. R. Schaum's Outline of Theory and Problems of College Algebra, 2nd ed. New York: McGraw-Hill, 1997.Uspensky, J. V. Theory of Equations. New York: McGraw-Hill, 1948.van der Waerden, B. L. Algebra, Vol. 2. New York: Springer-Verlag, 1991.van der Waerden, B. L. Geometry and Algebra in Ancient Civilizations. New York: Springer-Verlag, 1983.van der Waerden, B. L. A History of Algebra: From al-Khwārizmī to Emmy Noether. New York: Springer-Verlag, 1985.Varadarajan, V. S. Algebra in Ancient and Modern Times. Providence, RI: Amer. Math. Soc., 1998.Weisstein, E. W. "Books about Algebra." http://www.ericweisstein.com/encyclopedias/books/Algebra.html.

Algebra

## Cite this as:

Renze, John and Weisstein, Eric W. "Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Algebra.html