21-374 Field Theory - Spring 2004 MWF 1:30PM DH 1211

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 and algebraic number theory found 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 number theoretic techniques.

The basic ideas and methods required to study finite fields will also 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. price comparison.

Test Dates:

Will be held on Monday March first.

Evaluation: There will be two one hour tests (in class), weekly homework assignments, and a three hour final. These will be weighted as follows:

The standards of academic honesty as stated in the Student Handbook will be strictly enforced.

Prerequisites. Algebraic Structures or Math Studies.

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Last modified: February 18th, 2004