# Math 521A: Abstract Algebra

Course number: 21884
Fall 2010
MWF, 10:00 -10:50.
GMCS 328
San Diego State University
Final Exam: Fri. Dec. 17, 08:00-10:00

Our last day of class: Fri. Dec. 10

Professor: Mike O'Sullivan
Email: m.osullivan@math.sdsu.edu
Office: GMCS #579, 594-6697
Office Hours: MW 1:00-3:00,
You may also make an appointment for another time or stop by my office. If I am in and available, we can talk.

## Required Materials

Hungerford, Abstract Algebra: An Introduction 2nd ed.
Sage: open source mathematics software
Sage tutorial (9/20).

Review for the first exam.
Review for the second exam.
Review for the third exam.
Review for the final exam.

## Course Description

You are familiar with the term "algebra" from several years of math classes, dating back to elementary school. In this course, we will study many things from algebra that are familiar--factoring, fractions, functions--but we will explore deeper questions than you have seen in the past. In particular, we will *define* terms precisely, we will state *theorems* and we will *prove* them. This concern with *truth* makes this a course in pure mathematics. Since we start from simple assumptions and build theory from that basis, this mathematics is also called abstract.

We are using a well written text book in this course. It takes an unusual approach to the subject, starting with structures that you know well--the integers and polynomials--then developing the more general and abstract theory of rings and, finally, groups. The author organizes the material to emphasize central themes in abstract algebra: divisibility, the notion of congruence, the creation of new algebraic structures from known ones and homomorphims (functions that preserve structure). Chapters 1 and 2 discuss these notions for the integers; Chapters 4 and 5 discuss them for polynomial rings. Chapters 3, and 6 develop the same ideas for general rings. Chapter 7 is a new beginning, introducing the more general and abstract theory of groups. We will only cover the first few sections of that chapter.

This year I am introducing something new: the use of a computer software package called sage. This will allow us to compute examples that would be difficult (or even impossible) to do by hand, explore algorithms, and learn a bit of programming.

## Prerequisites

The main prerequisite is Discrete Mathematics, Math 245. It will be worthwhile to review the fundamentals of logic ("and", "or", "implies", "for all", "there exists", and negation). We will use a variety of proof methods in this course: direct proof, proof by contradiction, and mathematical induction. We use equivalence relations.

## Schedule

Here is a rough idea of the amount of time I expect to spend on each topic.

 SECTIONS TOPICS TIME Ch. 1,2 The integers: divisibility, primality. Congruence classes and arithmetic, the rings Z_m 3 weeks Ch, 3 General rings, and homorphisms 3 weeks Ch. 4 Polynomial rings, divisibility and irreducibility. 3 weeks Ch. 5 Congruence classes and arithmetic in polynomial rings. 1 week Ch. 6 Congruences classes and arithmetic for general rings. 2 weeks §7.1-5 Groups. 3 weeks

Written assignments should be carefully and neatly presented. You may work with others to solve problems, but the final written work should be done individually.

The relative weights of the work are given below.

 Weekly work 250 Tests 450 Final 300 Total 1000