math-ua.0343 / V63.0343: AlGEBRA I

Term:	Fall 2011
Meeting times:	TR 11:00 A.M. - 12:15 P.M.
Instructor:	Prof. Jonathan Goodman
Office:	TBA
Office hours:	TBA
Phone:	212-998-3326
Email:	goodman@cims.nyu.edu
Website:	http://www.math.nyu.edu/faculty/goodman/

Prerequisites

Students who wish to enroll in Algebra I must meet the following prerequisites:

Calculus III, V63.0123 with a grade of C or better

and

Linear Algebra, V63.0140 with a grade of C or better and/or the equivalent.
Additionally, it is suggested for students to have taken Analysis I V63.0325 as a prerequisite.

Goals and Topics

This is a first class in abstract algebra for undergraduate math majors or others interested in high level undergraduate algebra. The class will cover the basic topics of abstract groups and rings, but with many concrete examples. Much of the discussion of groups will use groups of matrices as examples. In particular, we will discuss linear algebra in finite fields, which are used now in cryptography and coding. Examples of rings include rings of algebraic integers and polynomials.

Much of the class is devoted to formulating axioms that capture the algebraic structures common to important examples. There is heavy emphasis on abstract reasoning and mathematical proof. There will be weekly homework assignments that involve writing proofs and working out examples.

The core of the class will be chapters 1-6 (groups) and 10-11 (rings) in the text. We will cover as much of the interesting material in the middle as time permits.

Course Details

Textbook and Materials

Algebra (second edition) by Michael Artin

Homework, Exams & Grading

The grade will be based on weekly homework assignments as well as one midterm and a final exam. The final exam is scheduled for Wednesday, May 11th from 4:00 tp 5:50 p.m. Location is TBD.

Tentative Calendar

Week	Topic
1	Chapter 1, Review of matrix operations from linear algebra
2	Sections 2.1 -- 2.6, Basic definitions of groups, subgroups, and mappings
3	Sections 2.7 -- 2.12, Product groups, equivalence classes, quotient groups, and modular arithmetic
4	Chapter 3, Abstract fields and vector spaces over abstract fields, examples from finite fields
5	Chapter 4, Matrices, eigenvalues, eigenvectors, and linear mappings of vector spaces
6	Chapter 6, Symmetries, symmetry groups, two dimensional crystallographic groups, symmetry groups of polygons and polyhedra, the counting formula
7	Sections 7.1 -- 7.5, Group actions, the icosahedral group, the cycle structure of permutations
8	Sections 7.6 -- 7.10, The Sylow theorems, groups of order 12, generators and relations
9	Sections 11.1 -- 11.3, Rings, mappings of rings, ideals, examples
10	Sections 11.4 -- 11.8, Quotient rings, adjoining elements, integral domains and fraction fields, the relation between ideals and geometry (briefly)
11	Sections 12.1 -- 12.3, Unique and non-unique factorization, primes, simple examples algebraic integers and polynomials, the Gauss lemma
12	Sections 12.4 -- 13.5, Examples of primes in integers in quadratic fields, the relation between factoring and ideals.
13	tbd