Math 305,
Modern Abstract Algebra I
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: 206
Distribution: Mathematical Modeling
