### Logistics

Class Room: Lockett 232

Time: TR 10:30am-11:50am

Office Hours: TR 12:00pm-1:30pm

### Resources

Textbooks

D. Dummit and R. Foote, Abstract Algebra, John Wiley & Sons, Inc., Hoboken, NJ.

P. Grillet, Abstract algebra, Graduate Texts in Mathematics, 242. Springer, New York 2007.

T.W. Hungerford, Algebra, Graduate Texts in Mathematics, 73. Springer-Verlag, New York, 1974.

L. Serge, Algebra, Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 1994.

P. Aluffi, Algebra: Chapter 0, Graduate Studies in Mathematics, 104. American Mathematical Society, Providence, RI, 2009

Qualifying Exam

### Week 1

Practice Qualifying Exam: Algebra Qualifying Exam Spring 2022

Practice Qualifying Exam Solutions: Algebra Qualifying Exam Solutions for Spring 2022

Note that there may be typos and mistakes in the above solutions. Please let me know if you find any, so they can be corrected.

Homework

Review Group Actions - Dummit and Foote Ch 4.1 - 4.4

Review Sylow Theory - Dummit and Foote Ch 4.5 - 4.6

Review Misc. Topics - Dummit and Foote Ch 2.2, Ch 6

Go over Practice Qualifying Exam Solutions

### Week 2 - Group Theory

Tuesday May 30th - Basic Group Theory

We will spend Tuesday working on basic group theory problems that use basic facts and ideas. These problems don't need to use Sylow theory or other advanced topics. Any leftover problems not covered in class are left as exercises that should be attempted without help from others. If you would like them graded, then you can hand them in during class, put them in my mailbox, or email me them at any time.

Problems: Basic Group Theory Problems

Solutions: Basic Group Theory Problem Solutions

Thursday June 1st - Sylow Theory and Less Basic Group Theory

We will spend Thursday working on less basic group theory problems which will use some ideas from ring theory and most notably Sylow theory. I have included some problems that have been included in prior qualifying exams, which are a bit harder than the rest. I have started to include a "problem difficulty" at the beginning of each problem. If a problem is listed as HARD it does not mean it is actually hard. It simply means that it often requires getting your hands dirty. None of these problems require any novel ideas, but some times you have to be ready to perform some calculations that other problems do not need. Likewise, a problem listed as EASY may not be very easy if you are not well prepared. It simply means the ideas presented are very routine and often elegant proofs exist that can be found quite quickly. If you are struggling with even the easy problems, do not fret. Just continue to work on the ideas and reach out if you need additional help.

Problems: Less Basic Group Theory Problems

Solutions: Less Basic Group Theory Problem Solutions

Review for Next Week: Make sure to review the basic definitions and ideas from ring theory including ideals, ring homomorphisms. It is also good to review the various types of rings like fields, PIDs, EDs, integral domains, etc. and their key theorems and properties. Finally, make sure to review the (undeniable) best examples of rings-- polynomial rings and their quotients!

### Week 3 - Ring Theory

Tuesday June 6th - Ring Theory I

We will spend Tuesday working on problems in ring theory centered around commutative rings. There is much more structure with commutative rings so the problems ten to be more involved (there are simply more options of things to do) then the group theory material. Unlike group theory as well is that every result is roughly the same difficulty, and many of these results can be greatly motivated by one's intuition on polynomials. As a result, the first day of class this week will center more on easy problems, rather than easy methods. This will hopefully provide a warm up for some of the more difficult arguments that we will attempt to tackle on Thursday.

Problems: Ring Theory Problems - Part I

Solutions: Ring Theory Problem Solutions - Part I

Thursday June 8th - Ring Theory II

I intended to make Part II of our ring theory problems more difficult, but in reality, there difference in difficulty with the problems is fairly minimal. The problems included in this section highlight many general ideas in ring theory and will hopefully illustrate many of the usual calculations one has to do to study rings. The problems ultimately are not that involved. Very rarely does one have to use a powerful and mysterious theorem. This differs quite a bit from group theory where the Sylow theorems and theorems revolving around group actions frequently have to be employed.

Problems: Ring Theory Problems - Part II

Solutions: Ring Theory Problem Solutions - Part II

### Week 4 - Module Theory

There are shockingly few problems on the algebra qualifying exam problem bank on modules (only 21, and we have already done one of them). As a result, I will assign (roughly) the first half on Tuesday and the second half on Thursday. This will ultimately give you solutions to all of the module problems in the qual bank. Module theory is still very important to cover for the qualifying exam. There is usually at least one problem on the exam, and they generally do not require any deep insights or even many computations. They are also the foundation for discussing linear algebra, which we will spend some considerable time on. Modules are the foundation for which a considerable amount of mathematics is developed: commutative algebra, algebraic geometry, homological algebra, sheaf theory, and representation theory. The problems also tend to be a lot of fun (in my opinion).

Tuesday June 13th - Module Theory I

We will mostly go through the first 11 problems in the module theory section of the qual bank, with one exception. I have deferred M6 until Thursday since it will require a more in-depth discussion of linear algebra then I wish to provide. The rest of the problems are quite manageable. I expect we will able to get through many more problems than usual. We will also emphasize 3 key classes of modules that will give a great foundation and make modules more intuitive. Roughly,

Modules over a PID <-(Like)-> Abelian groups

Free Modules <-(Like)-> Vector spaces

Ideals I and quotients R/I of rings R are an important example of R-modules

Problems: Module Theory Problems - Part I

Solutions: Module Theory Problem Solutions - Part I

Tuesday June 13th - Module Theory II

We will finish the remaining problems in the module theory section of the qual bank as well as covering M6 which we deferred. We already did M21 on the second day of class (it was in the practice qual), so it will be omitted. There are a fair few problems in this section the focus on elementary divisors, invariant factors, and general linear reduction of relations in abelian groups. These can be complicated and intimidating at first, but they are really quite approachable and the ideas come from the rich theory of linear algebra and the fundamental theorem of abelian groups.

Problems: Module Theory Problems - Part II

Solutions: Module Theory Problem Solutions - Part II

### Week 5 - Linear Algebra and Misc.

There are a lot of linear algebra problems in the algebra qual bank, but realistically, there will only be one problem (at most) that is solely linear algebra on the actual qualifying exam. While these problems are certainly foundational and extremely important for a student interested in anything related to algebra (algebraic geometry, number theory, or representation theory), there is not a large emphasis on these for the actual exam. We will at the least spend Tuesday doing some linear algebra problems, but it is likely we spend Thursday on other topics. This will ultimately be up to the class. The main concepts of interest are eigenvalues, eigenvectors, Jordan canonical forms, and a few problems that use inner products/adjoints. For the most part, these are all covered in a first class on linear algebra, with the exception of maybe Jordan canonical forms. The problems do tend to be proof based, which you may not have seen before. The biggest way these problems differ from your standard linear algebra problem taken as a freshman/sophomore is the use of non-real/non-complex fields. These add a layer of subtle complexity. Some of the problems will highlight how import algebraic closure of a field is, and we will see some weird behavior when your field is not algebraically closed. Nonetheless, much of this intuition can come from considering the reals and the complex numbers.

Tuesday June 20th - Linear Algebra I

We will do a selection of problems from the linear algebra section of the qual bank. The problems tend to way heavily on eigenvalues/eigenvectors and most importantly Jordan canonical forms. We will discuss some of the incredible power that Jordan canonical forms give and how fun they can be to compute. Depending on the interests of the class, this might be the only class we spend on linear algebra. To compensate, I have added a couple more problems than usual to the problem set.

Problems: Linear Algebra Problems - Part I

Solutions: Linear Algebra Problem Solutions - Part I

Thursday June 22nd - Ring Theory III

By popular demand, we will spend the Thursday on more problems from ring theory. We have already covered all of the basic ideas in ring theory, but there are still problems in the qual bank we have not covered. With the problem set for Thursday, we will have ~75% of the ring theory problems included. There are no new key ideas that are needed for these problems, and hopefully with 20 ring theory problems already completed, this should be a bit easier than before. We will also open the floor to any of the problems already covered in the Ring Theory I and Ring Theory II problem sets.

Problems: Ring Theory Problems - Part III

Solutions: Ring Theory Problem Solutions - Part III

### Week 6 - Rings, Modules, and Whatever Else

This is the final week for our short course in preparing for the algebra qualifying exam! We will finish off the course with some remaining problems that have not been covered from ring theory, module theory, maybe some group theory and linear algebra... The contents of this week will be a little sporadic and left mostly to the class. If you have suggestions on any topics you would like to see, please let me know! If we have time, we may even do another practice qualifying exam. I will also compile every single problem solution we have done throughout the semester into a concise document that will match the numbers of the qual bank. This will be a very useful (and dangerous) document. I urge you to use it only after you make a sincere and honest attempt at a problem. You should not look up a solution unless you absolutely must!

During late July or the second week of August, we will most likely hold a final few hour long review session. This will be entirely student-led, but I will serve as a moderator of sorts to keep things on track and to answer any problems if the students get stuck. I will update you all with a date when we are closer to qual time.

Also note that throughout the summer if you would like any of your solutions to be reviewed/graded, please feel free to email them to me and I would be happy to take a look. I can also continue to hold office hours by appointment during the remainder of the summer. So if you have any questions, please let me know.

Update: By popular demand, we will spend Tuesday working on the "hard" problems from the already existing problem sets. It is good to note that none of these problems are actually hard. Some require a trick that is a little unfamiliar and others may require a fairly tedious computation. Nonetheless, the ideas are quite straightforward. It is unlikely that more than one or two of these types of problems would appear on your qual. Although, it is still very important to preapre for them!

Tuesday June 27th - Hard Problem Compilation

Problems: Hard Problem Compilation

Solutions: Hard Problem Compilation Solutions

Thursday June 29th - Misc Problems in Modules and Linear Algebra

Problems: Misc Problems