Math 113

Abstract Algebra.

Homework.

Announcements.

Midterm Solutions.

General Policy.

Homework: There will be weekly homework. You may work with others, but you should write up your own solutions independently. Late homework will not be accepted. However, the lowest homework grade will not be counted.

Midterm:  There  will be one midterm on Thursday July 25th at regular class time.

Final Exam: The final exam will be on Thursday August 15th, at regular class time.
It is the policy of the Department of Mathematics that the following rules apply to final exams in all undergraduate mathematics courses: 1. The final exam must occur at the time and place designated on the College Final Exam Schedule. Instructors are not permitted to excuse students form the scheduled time of the final exam except in the cases of an Incomplete.

The final grade final grade will be 20% from the Homework, 30% from the midterm and 50% from the final.

Syllabus

• Preliminaries. We will begin with a review of some essential preliminaries, including sets, functions, relations, induction, and some very basic number theory. You have probably already seen this material in Math 55 or elsewhere, so the review will be brief. Some of this material is in section 0 of the book, some is scattered throughout random later sections, some is in the notes below on proofs, and some is in none of the above.
• Groups. We will learn a lot about groups, starting with the detailed study of a slew of examples, and then proceeding to some important general principles. We will cover most of Parts I, II, and III of the book.
  
• Ring theory and polynomials. Next we will learn about rings. We will pay particular attention to rings of polynomials, which are very important e.g. in algebraic geometry. We will cover most of Parts IV, V, and IX.
• Elements of field theory. Finally, after reviewing some notions from linear algebra in a more general setting, we will learn the basics of fields, from Part VI of the book. We will develop enough machinery to prove that one cannot trisect a sixty degree angle with a ruler and compass.
  

Course goals

A group is, roughly, a set with one "binary operation" on it satisfying certain axioms which we will learn about. Examples of groups include the integers with the operation of addition, the nonzero real numbers with the operation of multiplication, and the invertible n by n matrices with the operation of matrix multiplication. But groups arise in many other diverse ways. For example, the symmetries of an object in space naturally comprise a group. The moves that one can do on Rubik's cube comprise a fun example of a group. After studying many examples of groups, we will develop some general theory which concerns the basic principles underlying all groups.

A ring is, roughly, a set with two binary operations on it satisfying certain properties which we will learn about. An example is the integers with the operations of addition and multiplication. Another example is the ring of polynomials. A field is a ring with certain additional nice properties. At the end of the course we will have built up enough machinery to prove that one cannot trisect a sixty degree angle using a ruler and compass.

In addition to the specific topics we will study, which lie at the foundations of much of higher mathematics, an important goal of the course is to expand facility with mathematical reasoning and proofs in general, as a transition to more advanced mathematics courses, and for logical thinking outside of mathematics as well. I am hoping that you already have some familiarity with proofs from Math 55 or a similar course. If not, the following might be helpful:

Notes on proofs

Textbook

Most of the lectures will correspond to particular sections of the book (indicated in the syllabus below), and studying these sections should be very helpful for understanding the material. However, please note that in class I will often present material in a different order or from a different perspective than that of the book. We will also occasionally discuss topics which are not in the book at all. Thus it is important to attend class and, since you shouldn't expect to understand everything right away, to take good notes.

There are many other algebra texts out there, and you might try browsing through these for some additional perspectives. (Bear in mind that Fraleigh is an "entry-level" text, so many other algebra books will be too hard at this point; but after this course you should be prepared to start exploring these. There is a vast world of algebra out there!)

In addition, the math articles on wikipedia have gotten a lot better than they used to be, and much useful information related to this course can be found there. However you shouldn't blindly trust anything you read on the internet, and keep in mind that wikipedia articles tend to give brief summaries rather than the detailed explanations that are needed for proper understanding.

Study tips (for any upper division math course)

• It is essential to thoroughly learn the definitions of the concepts we will be studying. You don't have to memorize the exact wording given in class or in the book, but you do need to remember all the little clauses and conditions. If you don't know exactly what a UFD is, then you have no hope of proving that something is or is not a UFD. In addition, learning a definition means not just being able to recite the definition from memory, but also having an intuitive idea of what the definition means, knowing some examples and non-examples, and having some practical skill in working with the definition in mathematical arguments.
• In the same way it is necessary to learn the statements of the theorems that we will be proving.
• It is not necessary to memorize the proofs of theorems. However the more proofs you understand, the better your command of the material will be. When you study a proof, a useful aid to memory and understanding is to try to summarize the key ideas of the proof in a sentence or two. If you can't do this, then you probably don't yet really understand the proof.
  
• The material in this course is cumulative and gets somewhat harder as it goes along, so it is essential that you do not fall behind.
• If you want to really understand the material, the key is to ask your own questions. Can I find a good example of this? Is that hypothesis in that theorem really necessary? What happens if I drop it? Can I find a different proof using this other strategy? Does that other theorem have a generalization to the noncommutative case? Does this property imply that property, and if not, can I find a counterexample? Why is that condition in that definition there? What if I change it this way? This reminds me of something I saw in linear algebra; is there a direct connection?

Homework policy

When preparing your homework, please keep the following in mind:

1) You are encouraged to discuss the homework problems with your classmates. Mathematics can be a fun social activity! Perhaps the best way to learn is to think hard about a problem on your own until you get really stuck or solve it, then ask someone else how they thought about it. However, when it comes time to write down your solutions to hand in, you must do this by yourself, in your own words, without looking at someone else's paper.

2) All answers should be written in complete, grammatically correct English sentences which explain the logic of what you are doing, with mathematical symbols and equations interspersed as appropriate. For example, instead of writing "x^2 = 4, x = 2, x = -2", write "since x^2 = 4, it follows that x = 2 or x = -2." Otherwise your proof will be unreadable and will not receive credit. Results of calculations and answers to true/false questions etc. should always be justified. Proofs should be complete and detailed. The proofs in the book provide good models; but when in doubt, explain more details. Avoid phrases such as "it is easy to see that"; often what follows such a phrase is actually a tricky point that needs justifiction, or even false. You can of course cite theorems that we have already proved in class or from the book.