Math 28b: Introduction to Rings and Fields Spring 2003

Time: Monday, Wednesday, Thursday 1-2 pm

Room: 100 Goldsmith Hall

Instructor: Daniel Ruberman

Email: ruberman@brandeis.edu

Phone: x63074

Office: 310 Goldsmith Hall

Office Hours: Tuesday 10:30-12, Thursday 10:30-12

Homework: The homework assignments are in the following table.

 Date Due Assignment Thursday, Jan. 23 Homework #1 Thursday, Feb. 6 Homework #2 Wednesday, Feb. 19 Homework #3 Monday, Mar. 10 Homework #4 Wednesday, Mar. 26 Homework #5 Wednesday, Apr. 3 Homework #6 Monday, Apr. 14 Homework #7 Wednesday, Apr. 30 Homework #8

Course Description: This course is about two basic kinds of algebraic structures, rings and fields. Rings are sets of objects that can be added and multiplied, as we do with integers or polynomials. Fields are special kinds of rings, in which you can do division as well. Studying these mathematical objects gives insight into problems about polynomials and solutions of equations, and provides the basic language for number theory. We will begin with the most important examples: integers, rational numbers, complex numbers, Z/n, the Gaussian integers Z[i], and eventually study other examples such as polynomial rings, matrix rings, and quadratic extensions of the integers.

Fundamental concepts such as unique factorization, subrings, homomorphisms, and ideals will be introduced and explored. There are lots of applications of this material, and we will pick and choose from among these topics, depending on the interests of the students: elementary number theory, extension fields and constructibility of polygons, algebraic coding theory, etc.

Prerequisites: Math 23b and either Math 15a, 21a and b, 22a and b, or permission of the instructor. Note that Math 28b is independent of Math 28a, and so Math 28a is not a prerequisite.

Book: Contemporary Abstract Algebra (5th edition), by Joseph A. Gallian. Houghton-Mifflin, 1998. Available in the bookstore.

Coursework: There will be homework assignments, mainly problems from the book. These will be corrected and graded. There will be a midterm, and a final exam.