Abstract algebra

Abstract algebra is a part of math which studies algebraic structures. These include:

• groups
• rings
• fields
• matrices
• vector spaces
• algebras
• Boolean algebra

It is normal to build a theory on one kind of structure, like group theory or category theory.

The purpose of the theory of each concept is to organize the precise definition of the concept, examples of it, its substructures, the ways to relate different examples of the concept algebraically (these are called morphisms in some cases), and the concept's applications, both inside its own theory and outside in other areas of mathematics.

During history, different fields of mathematics have used algebras. Algebras are about finding or specifying rules on how to calculate with certain mathematical formulas and expressions. Another algebra (which is not abstract) is elementary algebra, for example.

Examples

• Solving equations with many variables. This leads to matrices, determinants and linear algebra.
• Finding formulas for polynomial equations. This led to the discovery of groups, as an expression of symmetry.
• Quadratic and higher-degree equations and Diophantine equations - especially when Fermat's last theorem was proved led to the definition of rings, and ideals.

Abstract Algebra Media

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  The permutations of the Rubik's Cube form a group, a fundamental concept within abstract algebra.

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  Algebraic structures between magmas and groups. For example, monoids are semigroups with identity.

Sources

• Allenby, R. B. J. T. (1991). Rings, fields, and groups : an introduction to abstract algebra (2nd ed.). London: E. Arnold. ISBN 978-0-340-54440-2.
• Hartman, Philip (January 2002). Ordinary differential equations (2nd ed.). Philadelphia: Society for Industrial and Applied Mathematics. ISBN 978-0-89871-510-1.
• Burris, Stanley N.; Sankappanavar, H. P. (1999) [1981], A Course in Universal Algebra
• Gilbert, Jimmie (2005). Elements of modern algebra (6th ed.). Belmont, CA: Thomson Brooks/Cole. ISBN 978-0-534-40264-8.
• Lang, Serge (21 June 2005). Algebra (Revised Third ed.). New York. ISBN 978-0-387-95385-4.
• Sethuraman, B. A. (1997). Rings, fields, and vector spaces : an introduction to abstract algebra via geometric constructability. New York: Springer. ISBN 978-0-387-94848-5.
• Whitehead, C. (6 December 2002). Guide to abstract algebra (2nd ed.). Basingstoke: Palgrave. ISBN 978-0-333-79447-0.
• Nicholson, W. Keith (20 March 2012). Introduction to abstract algebra (4th ed.). Hoboken: John Wiley & Sons. ISBN 978-1-118-13535-8.
• John R. Durbin (1992) Modern Algebra : an introduction, John Wiley & Sons