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Math 404: Modern Algebra, I Fall Semester 2007 Bradley University

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General information

Instructor: Dr. Alberto L. Delgado
Office: BR 466
Office Hours: MW 1:15-2:30, Th 11:30 - 12:00 and by appointment
Office Phone: 677-2504
Email: delgado@bradley.edu
Webpage: hilltop.bradley.edu/~delgado/index.html

The Course 

What is algebra?  What makes it modern?

Mathematics probably started with human's need to count: people, rocks, sheep.  Attempts to count various types of objects gave rise first to whole numbers, then to fractions.  Humans were eventually lead to the need to measure, especially physical properties: distance, temperature, time.  For these purposes, whole numbers and fractions weren't suitable; for example, the attempt to measure the length of a diagonal of a unit square gave rise to the need for irrational numbers.  The real numbers and, later, the complex numbers proved to be unavoidable as the need to make successively more complicated measurements arose.  What gives these various systems of measurement their lasting power is the operations -- addition, subtraction, multiplication, division -- which transform their objects and give them meaning.  These objects, these "numbers", provide the mathematical language; the operations, their grammar.   

As we move even further afield, we find new objects we want to measure: the growth of a population, the shape of a bell.  In order to deal with these, new systems of measurement have to be invented.  The systems which have been found most useful have one feature in common:  they all come with transforming operations. 

We can now answer the question we posed above: Modern Algebra is the branch of mathematics that deals with objects and the operations that transform them and imbue them with meaning.  In this course, we study the familiar objects of algebra -- the integers, fractions, real and imaginary numbers -- together with their familiar and less familiar operations.  We then abstract the data we glean from these objects and operations and investigate the concepts of groups, rings and fields.  

This course will also serve as an introduction to proofs and proof techniques.  You will learn about direct and indirect proofs; proof by contradiction and contrapositive; and proof by induction.  

Course Materials

The text for this course is A First Course in Abstract Algebra, third edition, by Joseph Rotman.  The text is available at the bookstore.  The text is not required for the course though. It contains all the material we will cover in the class, in approximately the same order as we will follow.  You will need to have some book where you can look things up, and this is an excellent text.  However there are others you could use -- some are more concise, some less so; some more complete, some less so.  I'll distribute a list of other books which you might want to look at.  Throughout the semester I'll make available to you various supplemental materials.  You will be able to download these from the webpage at the address above.

Prerequisites

You must have passed Math 207, Elementary Linear Algebra with Applications and Math 223, Calculus III  or their equivalent number somewhere else in order to take this course.  You must be prepared to commit substantial time and energy to this course.  Do not take this course unless you are ready to dedicate yourself to it fully.   

Grading

Your grade in the class will be based on your performance on homework assignments, two in-class examinations, and a final examination. The most important part of this course is the homework. Accordingly, there will be nearly weekly assignments. Your homework grade will make up 50% of your grade. The in-class examinations and the final together will make up the other 50% of your grade. The final exam is scheduled for 

Wednesday, 19 December from 9:00 to 11:00 p.m.