Math 113: Introduction to Abstract Algebra

Spring 2015

Instructor: David Li-Bland

Email:

GSI: Elan Bechor

Lectures: MWF, 3-4pm, 385 LeConte Hall

Course Control Number: 54155

Prerequisites: 54 or a course with equivalent linear algebra content.

Required Text: A First Course in Abstract Algebra, John Fraleigh (7th Edition)

Office Hours: Date: Time: Location:
David Li-Bland Monday 9am-12pm 1071 Evans Hall
Elan Bechor Tuesday 9am-12pm 853 Evans Hall
Elan Bechor Wednesday 9am-1pm 853 Evans Hall
Elan Bechor Thursday 9am-12pm 853 Evans Hall

Course Goals: Abstract algebra studies the structure of sets with operations on them. It aims to find general underlying principles common to the usual operations (addition, multiplication, etc.) on diverse sets such as integers, polynomials, matrices, permutations, and much more. We'll learn in particular about groups, rings, and fields.

A group is a set equipped with a binary operation satisfying certain axioms. Examples of groups include the integers with the operation of addition, or invertible n×n matrices with the operation of matrix multiplication, but there are many other examples (for instance, symmetries of geometric objects, permutations of finite sets, etc.). Thus, we will develop the general theory which describes common features of all groups.

A ring is a set equipped with two binary operations which satisfy certain properties similar to those of addition and multiplication. Examples of rings include integers, polynomials, or n×n matrices. A field is a ring with additional nice properties, such as the rational and real numbers.

In addition to these specific topics, an important goal of the course is to acquire more familiarity with abstract mathematical reasoning and proofs in general, as a transition to more advanced mathematical courses.  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:

Some notes of Michael Hutchings giving a very basic introduction to proofs are available here.

Some suggestions from Ian Agol for writing mathematics are available here.

Textbook

The textbook for this course is John B. Fraleigh, A first course in abstract algebra , 7th edition, Addison-Wesley. This book is very readable, has been well liked by students in the past, and contains lots of good exercises and examples.

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.

Suggested Readings will be posted here.

Equipment

If you do not already have a standard 3x3x3 rubik's cube at your disposal, please obtain one. The Rubik's cube will be used in a number of exercises.

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?
• If you get stuck on any of the above, you are welcome to come to my office hours. I am happy to discuss this stuff with you. Usually, the more thought you have put in beforehand, the more productive the discussion is likely to be.

Homework

With the exception of exam weeks, there will be weekly homework posted on the bCourses web page, which will be due in class each Friday. No late homework will be accepted.

A random selection of the assigned homework will be graded. Solutions to the graded exercises will be posted.

When calculating grades, we will drop your two lowest homework scores and use only your remaining scores.

1) You are encouraged to discuss the homework problems with your classmates. 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 to writing down your solutions, you must do this by yourself, in your own words, without looking at someone else's paper or any other source.

2) All answers should be written in complete 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." If your proof is unreadable it 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. You can of course cite theorems that we have already proved in class or from the book.

%20 Homework, %20 Midterm I, %20 Midterm II, %40 Final

Exams:

• Midterm Exam 1: Friday, February 13th
• Midterm Exam 2: Friday, March 20th.
• Final Exam: Wednesday, May 13, 2015 7-10pm

No make-up midterms will be given. If you are absent from one of the midterms, you must have a valid reason and present proof. In such a case, your final exam grade will be substituted for the missed midterm grade.

The L&S student calendar lists drop and grade-change deadlines. The deadline to drop is Friday, February 20th. The first midterm will be marked and returned to you on Wednesday, February 18th.

Syllabus

The following is a rough, tentative plan for the topics we will cover.

• 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 above notes 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. We will mostly skip the advanced group theory in Part VII, aside from stating a couple of the results. (You can learn some of this material in Math 114.) We will completely skip Part VIII on group theory in topology; this material is best learned in a topology course such as Math 142.
• 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,  and some of parts 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. We will not have time for the more advanced field theory in Part X, including the insolvability of the quintic; this is covered in Math 114.

Exercises and Reading Assignments: Exercises and reading assignments will be posted on the bCourses website. I will expect you to have completed these before each class.

Piazza: To handle questions posed outside of class, we will be using Piazza, a free platform for instructors to efficiently manage out-of-class Q&A. On the class dashboard, students can post questions and collaborate Wikipedia-style to edit responses to these questions. Instructors can also answer questions, endorse student answers, and edit or delete any posted content. Instead of emailing me math questions, I encourage you to post them to Piazza.

You will be able to access Piazza via the bCourses webpage for this class.

Approach to work and classroom/Piazza interactions: