What is Abstract algebra?
'''Abstract algebra''' is a part of [[mathematics|math]] which studies [[algebraic structure]]s. These include: * [[group (mathematics)|groups]] * [[ring (mathematics)|rings]] * [[Field (mathematics)|fields]] * [[Matrix (mathematics)|matrices]] * [[vector space]]s * [[Algebraic structure|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 [[morphism]]s 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 [[algebra]]s. 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 [[equation]]s with many variables. This leads to [[Matrix (mathematics)|matrices]], [[determinant]]s 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 equation]]s - especially when [[Fermat's last theorem]] was [[proof|proved]] led to the definition of rings, and [[ideal (mathematics)|ideals]]. ==Sources== * {{cite book |last1=Allenby |first1=R. B. J. T. |title=Rings, fields, and groups : an introduction to abstract algebra |year=1991 |publisher=E. Arnold |location=London |isbn=978-0-340-54440-2 |edition=2nd}} * {{cite book |last1=Hartman |first1=Philip |title=Ordinary differential equations |date=January 2002 |publisher=Society for Industrial and Applied Mathematics |location=Philadelphia |isbn=978-0-89871-510-1 |edition=2nd}} * Burris, Stanley N.; Sankappanavar, H. P. (1999) [1981], A Course in Univer