This is technically the second semester of a year-long advanced undergraduate algebra course, Math 370-371, but is being given “out-of-sequence”, i.e., in the fall rather than in the spring.   Some of you may not have had Math 370 (or its equivalent) and some who have may not have studied it recently, so we will go slowly at the start. However, there is some basic material you should learn or review before the start of class, including the following:

• Basic linear algebra, including the concepts of vector space and linear transformation
• Basic matrix theory, including the concepts of eigenvector, eigenvalue, and their application to diagonalization of matrices.  (We won’t need this until several weeks into the semester, but I am not going to stop to review it.)
• Elementary group theory – the concept of a group, subgroup, homomorphism, isomorphism, automorphism, normal subgroup, and quotient group.  (We won’t use the Sylow theorems until the very end of the course; I will remind you of their statements when we need them.)

Our text will be “Algebra”, by Michael Artin,  Prentice Hall, ISBN 0-13-0004763-5

(This book is encyclopedic. It contains, of course, all the above material but in more detail than you will need to get started in our Math 371.  It is also expensive – see if you can get a gently used copy. )

CONTENT OF THE COURSE

Click here to go to the first segment of the course

 In this semester we will cover the following

• Chapter 10, “Rings”, Sections 1 to 7, but we will not dwell on Section 2.
• Chapter 11, “Factorization”, Sections 1 to 5.
• Chapter 12, “Modules”, Sections 1 to 7. My class presentation may be a little different from that in the book.  For example, it takes very little more effort to prove the structure theorem for modules over a principal domain (as I will do in class) than it does for a Euclidean domain (in the book).  If you find the book clearer than my exposition then make sure you understand what’s there; it will be sufficient.
• Chapter 13, “Fields”.
• Chapter 14,   “Galois Theory”.  My exposition of Galois theory in class will follow the very small volume “Galois Theory” by Emil Artin, father of Michael Artin.  The elder Artin,  in addition to being one of the great mathematicians of the past century,  was also one of its greatest expositors, and this little volume is an absolute gem. Everything it contains and much more is also in our text by his son, but the lucidity of this little treatise is unmatched. It is definitely worth having.  (Exercises, however, will be taken from the text.)

In the past I have sometimes omitted Galois Theory in favor of Chapter 9, “Group Representations”.  If there is time at the end of the semester I will try to present some of the basic ideas.  This is a useful topic for physicists. If we can not get to it in class I will work with anyone who wants to pursue it independently.

EXAMS AND GRADING

This is an ambitious program and class time is precious.  Therefore, all exams, including the final will be take-home. The exams will be difficult and sometimes may require you to learn new material.  You are encouraged to study and work in groups. At the start of the semester I will help you to organize.  Here is the point break-down:

• First mid-term:               25%
• Second mid-term:        25%
• Final:                                    35%
• Homework:                      15%

There will be no unannounced quizzes.

There will be very stringent rules for presentation of your work, particularly of exams.  Even though you may work in groups, both for exams and for homework, everyone must submit his or her own paper, in his or her own hand. (No machine copies!) For the exams, there must be a cover page with only your name, a list of problems you have attempted, and the names of all with whom you have worked.  (If it appears that everyone in a group has misunderstood a problem then I can get you together and explain.) Each answer must start on a separate page.  You may use outside sources with the following rules: You may consult other faculty members asking for hints about problems or sources to which you can go but must make it clear that this is part of your exam. You may not consult any of the graduate students. (The faculty members will limit themselves to hints or direct you to sources; graduate students tend to do too much.) You may use the web, and if you find a solution to one of my problems on the web you may use it, provided that (i) you give the source and (ii) you understand it.  (Whether you do or not will be evident from what you write.) Be neat! If a paper is too painful to read then I won’t read it.

RESOURCES AND SUGGESTIONS

I will put some elementary books on reserve at the beginning of the semester which may help those who are rusty to get up to speed. The web has also become a good resource. There are even complete courses in algebra available on the web -- it sometimes helps to see things explained in a different way.   Above all, keep up with the course.  If you do not understand something that I have said in class, stop me and ask a question!  I will almost always be available immediately after class and will arrange additional hours, if necessary, to meet your schedules.