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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.
- 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.