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Abstract algebra
Abstract algebra is a part of math which studies algebraic structures. These include:
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 higherdegree equations and Diophantine equations  especially when Fermat's last theorem was proved led to the definition of rings, and ideals.
Sources
 Allenby, R. B. J. T. (1991), Rings, Fields and Groups, ButterworthHeinemann, ISBN 9780340544402
 Artin, Michael (1991), Algebra, Prentice Hall, ISBN 9780898715101
 Burris, Stanley N.; Sankappanavar, H. P. (1999) [1981], A Course in Universal Algebra
 Gilbert, Jimmie; Gilbert, Linda (2005), Elements of Modern Algebra, Thomson Brooks/Cole, ISBN 9780534402648
 Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: SpringerVerlag, ISBN 9780387953854, MR 1878556
 Sethuraman, B. A. (1996), Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility, Berlin, New York: SpringerVerlag, ISBN 9780387948485
 Whitehead, C. (2002), Guide to Abstract Algebra (2nd ed.), Houndmills: Palgrave, ISBN 9780333794470
 W. Keith Nicholson (2012) Introduction to Abstract Algebra, 4th edition, John Wiley & Sons ISBN 9781118135358 .
 John R. Durbin (1992) Modern Algebra : an introduction, John Wiley & Sons

