Algebra

The branch of mathematics dealing with Group Theory and Coding Theory which studies number systems and operations within them. The word ``algebra'' is a distortion of the Arabic title of a treatise by al-Khwarizmi about algebraic methods. Note that mathematicians refer to the ``school algebra'' generally taught in middle and high school as ``Arithmetic,'' reserving the word ``algebra'' for the more advanced aspects of the subject.

Formally, an algebra is a Vector Space $V$, over a Field $F$ with a Multiplication which turns it into a Ring defined such that, if $f\in F$ and $x, y\in V$, then

$$f(xy) = (fx)y = x(fy).$$

In addition to the usual algebra of Real Numbers, there are $\approx 1151$ additional Consistent algebras which can be formulated by weakening the Field Axioms, at least 200 of which have been rigorously proven to be self-Consistent (Bell 1945).

Algebras which have been investigated and found to be of interest are usually named after one or more of their investigators. This practice leads to exotic-sounding (but unenlightening) names which algebraists frequently use with minimal or nonexistent explanation.

See also Alternate Algebra, Alternating Algebra, B*-Algebra, Banach Algebra, Boolean Algebra, Borel Sigma Algebra, C*-Algebra, Cayley Algebra, Clifford Algebra, Commutative Algebra, Exterior Algebra, Fundamental Theorem of Algebra, Graded Algebra, Grassmann Algebra, Hecke Algebra, Heyting Algebra, Homological Algebra, Hopf Algebra, Jordan Algebra, Lie Algebra, Linear Algebra, Measure Algebra, Nonassociative Algebra, Quaternion, Robbins Algebra, Schur Algebra, Semisimple Algebra, Sigma Algebra, Simple Algebra, Steenrod Algebra, von Neumann Algebra

References

Algebra

Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991.

Bell, E. T. The Development of Mathematics, 2nd ed. New York: McGraw-Hill, pp. 35-36, 1945.

Bhattacharya, P. B.; Jain, S. K.; and Nagpu, S. R. (Eds.). Basic Algebra, 2nd ed. New York: Cambridge University Press, 1994.

Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, 1996.

Brown, K. S. ``Algebra.'' http://www.seanet.com/~ksbrown/ialgebra.htm.

Cardano, G. Ars Magna or The Rules of Algebra. New York: Dover, 1993.

Chevalley, C. C. Introduction to the Theory of Algebraic Functions of One Variable. Providence, RI: Amer. Math. Soc., 1951.

Chrystal, G. Textbook of Algebra, 2 vols. New York: Dover, 1961.

Dickson, L. E. Algebras and Their Arithmetics. Chicago, IL: University of Chicago Press, 1923.

Dickson, L. E. Modern Algebraic Theories. Chicago, IL: H. Sanborn, 1926.

Edwards, H. M. Galois Theory, corrected 2nd printing. New York: Springer-Verlag, 1993.

Euler, L. Elements of Algebra. New York: Springer-Verlag, 1984.

Gallian, J. A. Contemporary Abstract Algebra, 3rd ed. Lexington, MA: D. C. Heath, 1994.

Grove, L. Algebra. New York: Academic Press, 1983.

Hall, H. S. and Knight, S. R. Higher Algebra, A Sequel to Elementary Algebra for Schools. London: Macmillan, 1960.

Harrison, M. A. ``The Number of Isomorphism Types of Finite Algebras.'' Proc. Amer. Math. Soc. 17, 735-737, 1966.

Herstein, I. N. Noncommutative Rings. Washington, DC: Math. Assoc. Amer., 1996.

Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 1975.

Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. Freeman, 1989.

Kaplansky, I. Fields and Rings, 2nd ed. Chicago, IL: University of Chicago Press, 1995.

Lang, S. Undergraduate Algebra, 2nd ed. New York: Springer-Verlag, 1990.

Uspensky, J. V. Theory of Equations. New York: McGraw-Hill, 1948.

van der Waerden, B. L. Algebra, Vol. 2. New York: Springer-Verlag, 1991.

van der Waerden, B. L. Geometry and Algebra in Ancient Civilizations. New York: Springer-Verlag, 1983.

van der Waerden, B. L. A History of Algebra: From al-Khwarizmi to Emmy Noether. New York: Springer-Verlag, 1985.

Varadarajan, V. S. Algebra in Ancient and Modern Times. Providence, RI: Amer. Math. Soc., 1998.