Math 305, Abstract Algebra

This course deals with algebraic structures that come up in many different areas of mathematics and science. Take a tetrahedron, for example. All the rotations of three-space which move the tetrahedron so that it occupies the same space that it started in (with vertices in possibly different positions) are called the symmetries of the tetrahedron. There is an operation on two symmetries which behaves (sort of) like ordinary multiplication of numbers: if A and B are two symmetries (rotations), then A*B can be defined to be the rotation applied to the tetrahedron by first rotating according to B and then rotating according to A. The interesting thing about this "multiplication" is that it is associative but not commutative. The set of all symmetries (and there are exactly 12 of them) thus forms an algebraic structure, which in this case is called a group.

The course starts from the axioms for a group and develops many of the beautiful properties of groups in detail. The other main structures studied in the course are rings, which are sets with an addition and a multiplication that have similar properties to those satisfied by the set Z of integers and the set R of real numbers. An important example is the set Q[x] of all polynomials with rational coefficients. Among other things, it is proved that these polynomials factor uniquely into "prime" (or "irreducible") polynomials, in the same way that integers factor uniquely into prime numbers. This has important applications to things like satellite communication and efficient telephone communication. Groups are very important in physics (quantum mechanics) and chemistry (study of bonding and spectra of molecules).

Prerequisite: Math 206.
Distribution: Mathematical Modeling.

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Created by: Heather Barrett '08
Maintained by: Professor Alexia Sontag
Date Created: June 2005
Last Modified: August 23, 2007
Expires: August, 2008