Helen Lei


Here are some eccentric files I made in the course of studying for the algebra exam. I was in a quandary in that I knew enough algebra to not want to read an algebra book but not enough algebra to pass the exam. Also, it was rather difficult to find a good algebra book to commit to and study out of.

So I pretended that I had learned algebra but then had a serious frontal lobotomy and forgot all of it (actually not so far from the truth). My idea was then to somehow take some sort of random walk through the subject and hope the walk is strongly recurrent. The walk was not quite as recurrent as I had hoped, but recurrent enough to enable me to pass the exam. The random walk took place in a three-dimensional sphere. The walker starts at the surface, with the goal of getting as close to the core as possible.

To guide myself through the walk, I decided to pick 4 old exams (Spring 2002, Winter 2002, Fall 2002, Winter 2003) and simply try to solve or find solutions to all the problems. After the solution to each problem was found, I would then list all relevant things related to the problem. The exam had four sections and I did group theory first, followed by field/galois theory, ring theory, and finally linear algebra.

Group Theory

In here you will find things like group action, Sylow's Theorems, basic properties of the symmetric group and free abelian group. I actually found A Course in Algebra by Vinberg to be useful here and elsewhere. The nice thing about this book is that some basic definitions are stated clearly and without too much clutter. The drawback is the index is horrible.

Galois Theory

In here you will find in rather jumbled order a lot of the things one learns in a first course in galois theory. In addition there's a section on semidirect products (useful sometimes for finding galois groups). I must admit I slightly dreaded this section since there's a lot one can do (think finite but really large number). A good book to read for the basic development is of course Artin's Galois Theory. For more involved/detailed things, look in Dummit and Foote.

Ring Theory

I don't think I did such a good job studying ring theory. There are a couple of interesting problems from Winter 2002 and Winter 2003. You will also find a section on tensor products.

Linear Algebra

By the time I got to linear algebra, I really did not have the time or patience to type up all the solutions, especially since they often involve matrices. Mainly what you will find here are a list of properties of the determinant function and more importantly, a description of Jordan and Rational Canonical forms and some immediate consequences.

Miscellaneous Things

I started a file to list additional things that came up in doing more old exams or looking through solutions that I either didn't know or didn't know well enough, but didn't get very far.