# Math 113: Abstract algebra

## UC Berkeley, Fall 2010

Section 2, T/Th 11:10-12:30
Section 4, T/Th 3:40-5

Announcement: Office hours are cancelled Wednesday, November 24
Announcement: There will be review on Thursday, December 9, same time & place as class.

## Instructor

Ian Agol
ianagol at math.berkeley.edu
Office phone: 510-642-4377.
Office: 921 Evans.
Office Hours: Monday 11:10-12 am, Wednesday 2:10-4 pm, or by call or e-mail to set an appointment

## Course goals

In previous courses you have seen many kinds of algebra, from the algebra of real and complex numbers, to polynomials, functions, vectors, and matrices. Abstract algebra (most mathematicians would just call this "algebra", I'm not sure why the word "abstract" is there) encompasses all of this and much more. Roughly speaking, abstract algebra studies the structure of sets with operations on them. We will study three basic kinds of "sets with operations on them", called Groups, Rings, and Fields.

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. 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, such as the rational and real numbers. 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

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

## 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.

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.

Other references:

Algebra: Abstract and Concrete, Edition 2.5, Frederick M. Goodman

Applied Abstract Algebra,  Rudolf Lidl and Gunter Pilz: this book gives applications of abstract algebra, but is a second course (you can read online through the library).

Basic Algebra, Groups, Rings and Fields, P. M. Cohn

Abstract Algebra, Paul Garrett (chapter 01 has some background on number theory)

## 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 policy

Homework is due every Thursday (except for the first two Thursdays and the weeks of the midterms or Thursday holidays) at the beginning of class. You can either bring it to class or slide it under my office door. (If it doesn't fit under the door, please be more concise!) Homework assignments will be posted below at least a week before they are due. No late homeworks will be accepted for any reason, so that we can go over the homework problems at the beginning of Thursday's class (which is when people are most eager to see solutions to troublesome problems). However it is OK if you miss the deadline once or twice, because your lowest two homework scores will be dropped.

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 copying or looking at someone else's paper. If you obtain help from another student with a problem, write this down on your homework. This will not affect your grade, unless it is clear that you have copied verbatim - the final answer should be written in your own words. It is important to get in the habit of citing your sources, which may include your colleagues (otherwise, you are plagiarizing)!

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.

## Reading Assignments (from Fraleigh)

• 8/31: Read Notes on proofs, Suggestions for writing mathematics, and Sections 0, 1 of Fraleigh
• 9/2: Read Sections 2 and 3
• 9/7: Read Sections 3 and 4
• 9/9: Read Sections 5 and 6
• 9/14: Read Sections 6 and 7
• 9/16: Read Sections 8 and 9
• 9/21: Read Sections 8 and 9
• 9/23: Read Sections 9 and 10
• 9/28: Read Sections 10 and 11
• 9/30: Read Sections 11 and 12
• 10/5: Read Section 12
• 10/7: Midterm 1
• 10/12: Read Sections 12 and 13
• 10/14: Read Section 13
• 10/19: Read Section 14
• 10/21: Read Section 15
• 10/26: Read Section 18
• 10/28: Read Section 19
• 11/2: Review
• 11/4: Midterm 2
• 11/9: Read Section 20
• 11/11: Veteran's day, no class
• 11/16: Read Section 21
• 11/18: Read Section 22
• 11/23: Read Section 23
• 11/25: Thanksgiving, no class
• 11/30: Read Section 26
• 12/2: Read Section 27

## Homework assignments:

• Homework 1 due 9/9 (at the beginning of class): Section 0, #29-34*; Section 2, #5,6,9,13,23,37, *explain for each problem from Section 0 which property of equivalence relations hold, and which do not, with justification.  Homework 1 solutions (cut and paste into a web browser): http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_1_sol.pdf
• Homework 2 due 9/16: Section 3 # 2, 4, 6,  27, 33 ; Section 4 # 9, 10, 37; Section 5 # 13, 23, 49; Section 6 # 22, 48. Homework 2 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_2_sol.pdf
• Homework 3 due 9/23: Section 7 # 11, 18, Section 8 # 3, 8,  21, 45, 46. Homework 3 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_3_sol.pdf
• Homweork 4 due 9/30: Section 9 # 7, 13, 16, 29, Section 10, # 6, 7, 28, 29, 39. Homework 4 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_4_sol.pdf
• Solutions to Midterm 1
• Homework 5 due 10/14: Section 11, # 10, 18, 20, 30, 47, Section 12 #8, 16, 24-30, 38, 40. Homework 5 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_5_sol.pdf
• Homework 6 due 10/21: Section 13, # 2, 6, 8, 20, 36, 37, 44, 50  Section 14 # 3, 6, 27, 31, 32,  33. Homework 6 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_6_sol.pdf
• Homework 7 due 10/28: Section 15, # 3, 7, 13, 36, 37, Section 18 # 9, 15, 20, 25, 37, 41. Homework 7 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_7_sol.pdf
• No homework due 11/11 because of Veteran's day
• Solutions to Midterm 2 Section 2: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_mt_2_Section_2_solutions.pdf
• Solutions to Midterm 2 Section 4: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_mt_2_Section_4_solutions.pdf
• Homework 8 due 11/18: Section 19,  #3, 9, 25, 29; Section 20 #6, 7, 14,  27, 28; Section 21 #1, 2, 5, 7. Homework 8 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_8_sol.pdf
• Homework 9 due 12/2: Section 22, # 5, 14, 25, 30; Section 23 #3, 12, 17, 28, 34; Section 26 #3, 17, 18, 20; Homework 9 solutions: http://dl.dropbox.com/u/8592391/HomeworkSolutionsMath113/113_Fall10_hw_9_sol.pdf

## Exams and grading

There will be in-class midterms on 10/7 and 11/4, and a final exam on 12/15, 8-11 am, 6 Evans (Section 2)  or  12/17, 7-10 pm, 71 Evans (Section 4).

There will be no makeup exams. However you can miss one midterm without penalty, as explained in the grading policy below.

There is no regrading unless there is an egregious error such as adding up the points incorrectly. Every effort is made to grade all exams according to the same standards, so regrading one student's exam would be unfair to everyone else.

The course grade will be determined as follows: homework 20%, midterms 20% each, final 60%, lowest exam score -20%. All grades will be curved to a uniform scale before being averaged.

## Syllabus (short version)

• 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 consider a few examples which are not in the book, such as symmetry groups of polyhedra and wallpaper groups. 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, 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.

• ## 8/26, 8/31: About the course. Brief review of preliminaries: sets, functions, injections, surjections, bijections. Equivalence relations and modular arithmetic. Proof by induction, strong induction, and the well ordering principle. The division theorem. Greatest common divisors, the Euclidean algorithm, and solving ax = b (mod n). The fundamental theorem of arithmetic. [0,notes on proofs]. For fun: Play the game Set

• 9/2: Binary operations. Isomorphism of binary structures. Using structural properties to show that binary structures are not isomorphic. [Sections 2,3]
• 9/7: Groups: definition, many examples [Section 4]
• 9/9: Subgroups, cyclic groups [Sections 4,5]
• 9/14: cyclic groups [Section 6]
• 9/16: Cayley graphs [Sections 7]
• 9/21: permutation groups [Sections 8, 9]
• 9/23: orbits, alternating groups, cosets [9,10]
• 9/28: Lagrange's theorem [10]
• 9/30: Product groups [11]
• 10/5: review
• 10/12: plane isometries,  frieze and wallpaper groups [12]  Wallpaper patterns
• 10/14: homomorphisms [13]
• 10/19: factor groups [14]
• 10/21: factor group computations and simple groups [15]
• 10/26: rings and fields [18]
• 10/28: integral domains [19]
• 11/9: Fermat's and Euler's theorems [20]
• 11/16: Field of quotients [21]
• 11/18: Polynomial rings [22]
• 11/23: Factorization of polynomials [23]
• 11/30: Homomorphisms and factor rings [26]
• 12/2: Prime and maximal ideals [27]