21-374 Field Theory - Spring 2019 MWF 11:30 , WEH 7201

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. price comparison.

Test Dates: The first midterm test will be held on Wednesday 2/27/2019 instead of a usual lecture. The date of the second midterm test will be announced.

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.

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Last modified: February 20th, 2019