Instructor: Rami Grossberg
Office: WEH 7204
Phone: x8482 (268-8482 from external lines), messages at
x2545
Email: Rami@cmu.edu
URL:
www.math.cmu.edu/~rami
Office Hours: Immediately after class or by appointment.
Purpose. Field theory has central importnace
in several branches of modern mathematics among them are: Number theory,
geometry and algebra. In recent years field theory
found an increasing role in theoretical computer
science especially in connections with complexity theory and cryptography.
The goal of this course is to provide a successor to
Algebraic Structures (21-373), with an emphasis on applications of
groups, rings, and fields within algebra to some major classical
problems. These include constructions with a ruler and compass, and
(un)Ęsolvability of equations by radicals. It also offers an opportunity
to see group theory and basic ring theory "in action", and introduces
several powerful tools for number theory and algebraic geometry.
One of the most important aspects of this course was development of the basics of Galois Theory, the
study of various groups
and connections to fields. Application will include using discrete mathematics of
the fundamental theorem of algebra and the structure of finite fields.
Another aspect will be the study of the structure of algebraically closed fields.
The basic ideas and methods required to study finite fields will be
introduced, these have recently been applied in a number of areas of
theoretical computer science including primality testing and
cryptography.
Course description. We will start with a review of ring theory.
Definitions and examples, field extensions, adjunction of roots,
algebraic numbers, dimension formula, constructions with ruler and
compass (it is impossible to trisect an arbitrary angle, and it is
impossible to duplicate the cube), splitting fields, existence (and
uniqueness) of algebraic closure, symmetric polynomials, Galois groups,
Galois extensions, the Galois correspondence theorem for characteristics
0, permutations and simplicity of An, unsolvability by radicals of the
general quintic, characterization of finite fields (and their
multiplicative groups), Wedderburn's theorem (optional), transcendental
extensions, Steinitz's theorem on trascendence degree.
Text: "Abstract Algebra" by D. S. Dummit & R. M. Foote. 3rd
edition Published by John Wiley & Sons, 2003.
Remote teaching comments. While HW will be assigned, the HW will be at times
not reflective of the material presented in class. It would be nearly impossible to be successful in this course
without following the lectures closely, doing the HW alone is insufficient.
While I have no interest policing people and I believe that people should be allowed as much freedom as possible. I would like to emphasize that
attendance
is not a formal requirement it is difficult to over emphasize the importance of attending the lectures.
Learning this material directly from the textbook
without attending the lectures would be very hard.
Test Dates: The dates of the of the tests will be announced.
Evaluation: There will be two one hour mid term tests (instead of a regular lecture), weekly homework
assignments, and a three hour final. These will be weighted as
follows:
Prerequisites. Algebraic Structures.
Rami's home page.
| Last modified: January 13th, 2019 |