What is Algebra?

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About 8 months ago I wrote about Analysis:

Thus, algebraic approximations produced the algebra of inequalities. The application of Algebra of inequalities lead to concept of Approximations in Calculus.

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You may have seen/heard this quote several times…

Now the time has come to understand the term “Algebra” itself, which has very rich history and dynamic present. I will use following classification (influenced by Shreeram Abhyankar) of algebra in 3 levels:

  1. High School Algebra (HSA)
  2. College Algebra (CA)
  3. University Algebra (UA)

HSA (8th Century – 16th Century) is all about learning tricks and manipulations to solve mensuration problems which involve solving linear, quadratic and “special” cubic equations for real (or rational) numbers. This level was developed by Muḥammad ibn Mūsā al-KhwārizmīThābit ibn QurraOmar KhayyámLeonardo Pisano (Fibonacci)Maestro Dardi of PisaScipione del FerroNiccolò Fontana (Tartaglia)Gerolamo CardanoLodovico Ferrari and Rafael Bombelli.

CA (18th Century – 19th Century) is commonly known as abstract algebra. Its development was motivated by the failure of HSA to solve the general equations of degree higher than the fourth and later on the study of symmetry of equations, geometric objects, etc. became one of the central topics of interest. In this we study properties of various algebraic structures like fields, linear spaces, groups, rings and modules. This level was developed by Joseph-Louis LagrangePaolo RuffiniPietro Abbati MarescottiNiels AbelÉvariste GaloisAugustin-Louis Cauchy Arthur CayleyLudwig SylowCamille JordanOtto HölderCarl Friedrich GaussLeonhard EulerWilliam Rowan Hamilton, Hermann GrassmannHeinrich Weber Emmy Noether and Abraham Fraenkel .

UA (19th Century – present) has derived its motivations from many diverse subjects of study in mathematics like Number Theory, Geometry and Analysis.  In this level of study, the term “algebra” itself has a different meaning

An algebra over a field is a vector space (a module over a field) equipped with a bilinear product.

and topics are named like Commutative Algebra, Lie  Algebra and so on. This level was initially developed by Benjamin Peirce,  Georg FrobeniusRichard DedekindKarl WeierstrassÉlie CartanTheodor MolienSophus LieJoseph WedderburnMax NoetherLeopold Kronecker,  David HilbertFrancis Macaulay,  Emanuel LaskerJames Joseph SylvesterPaul Gordan, Emil ArtinKurt HenselErnst SteinitzOtto Schreier ….

Since algebra happens to be a fast developing research area, the above classification is valid only for this moment. Also note that, though Emmy Noether was daughter of Max Noether I have included the contributions of Emmy in CA and those of Max in UA. The list of contributors is not exhaustive.

References:

[1] van der Waerden, B. L.  A history of algebra. Berlin and Heidelberg: Springer-Verlag, 1985. doi: 10.1007/978-3-642-51599-6

[2] Kleiner, I.  A History of Abstract Algebra. Boston : Birkhäuser, 2007. doi: 10.1007/978-0-8176-4685-1

[3] Burns, J. E. “The Foundation Period in the History of Group Theory.” American Mathematical Monthly 20, (1913), 141-148.  doi: 10.2307/2972411

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9 responses »

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  8. U have written a nice little precis or synopsis of history of algebra !! or outlines !! One could try to fill the gaps and make a book of history of algebra from your little article !! I do not know anything about level 3!

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