# Math 113: Abstract algebra

## Instructor

Michael Hutchings
hutching@math.berkeley.edu
Office phone: 510-642-4329.
Office: 923 Evans.
Tentative office hours: Wednesday 9:30 - 11:30.

## 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. 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

Some old notes of mine giving a very basic introduction to proofs 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.

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.

## Equipment

It is recommended that you obtain a 3x3 Rubik's cube, if you do not already have one, e.g. from rubiks.com. This will be used to illustrate some group-theoretic concepts in a fun, hands-on way. (However ability to solve Rubik's cube is not a course requirement. By the way, you can find some pretty scary stuff by searching for "rubik" on youtube.)

## 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. (When I was a student I had a deck of index cards; on each card I wrote the statement of a theorem on one side and a summary of the proof on the other side. Very useful!)
• 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 Wednesday (except for the first two Wednesdays and the weeks of the midterms) at 2:10 PM sharp . 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 Wednesday'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.

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.

## Homework assignments

There will be in-class midterms on 10/7 and 11/9, and a final exam on 12/17.

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 Rubik's cube and some symmetry 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.

## Syllabus (long version)

The following is the plan for what I intend to cover, and when. Below, numbers in square brackets refer to the relevant sections of Fraleigh. The schedule of topics is only an approximation, so some topics might be covered at slightly different times than listed, or skipped if time gets short.
• 8/26,8/28, 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].
• 9/2, 9/4: Binary operations. Isomorphism of binary structures. Using structural properties to show that binary structures are not isomorphic. [2,3] (We won't follow the rather discursive [1].)
• 9/7: NO CLASS (Labor Day holiday)
• 9/9, 9/11: Groups. Lots of examples (not all in the book): cyclic groups, Z_n^*, dihedral groups, the Klein 4-group, matrix groups, symmetry groups of polyhedra, permutation groups, the Rubik's cube group. [4]
• 9/14: Subgroups. Lots of examples. [5]
• 9/16: Cyclic groups and their subgroups. [6] (We will skip [7], except that we may say something about the subgroup of a group generated by a subset of the group.)
• 9/18, 9/21. All about permutations: Every permutation is a product of disjoint cycles (uniquely up to reordering the factors). Every permutation is a product of transpositions. Dichotomy between even and odd permutations. The alternating group. [8,9]
• 9/23, 9/25: Cosets, Lagrange's theorem. [10]
• 9/28: Direct products. Fundamental theorem of finitely generated abelian groups (statement only). [11] (We will skip [12].)
• 9/30: Homomorphisms. [13]
• 10/2: Conjugation. Normal subgroups. Quotient of a group by a normal subgroup. (Quotient groups are called "factor groups" in the book.) [14]
• 10/5: Review for the midterm.
• 10/7: MIDTERM #1, in class. Covers the material up to and including the lecture on 9/30.
• 10/9, 10/12: More about normal subgroups and quotient groups. The "fundamental homomorphism theorem". Commutator subgroup, abelianization, center. [14,15]
• 10/14: Group actions and Burnside's formula. [16,17]
• 10/16: Rings and fields. Basic definitions and lots of examples. [18]
• 10/19: Integral domains. [19] (We will skip [20]; we will have done most of this earlier.)
• 10/21: Quotient field of an integral domain. [21]
• 10/23: Rings of polynomials. Evaluation homomorphisms. If R is an integral domain then so is R[x]. [22]
• 10/26, 10/28, 10/30: Factorization of polynomials over a field: Division theorem for polynomials. Factor theorem. Multiplicative groups of finite fields are cyclic. Irreducible polynomials. Irreducibility over Z implies irreducibility over Q. (The book defers the proof from [23] to [45].) Eisenstein's crieterion. [23] Greatest common divisor and Euclidean algorithm in F[x] (cf [46]). Unique factorization in F[x], using the above. (We will skip [24,25].)
• 11/2: Ideals and quotient rings. [26]
• 11/4: Prime and maximal ideals. [27]
• 11/6: Review for the midterm.
• 11/9: MIDTERM #2, in class. Covers the material up to and including the lecture on 11/2, with emphasis on the material not covered on the first midterm.
• 11/11: NO CLASS (Veteran's Day holiday)
• 11/13: More about prime and maximal ideals. [27] (We will skip [28].)
• 11/16, 11/18: Euclidean domains, PID's, and UFD's. Fun with Gaussian integers. Any prime congruent to 1 mod 4 is the sum of two squares. (Selection of topics from [45,46,47].) (We might not have time for all of this, but it brings together a lot of what we have learned about rings so it is a good review.)
• 11/20, 11/23, 11/25: Field extensions, algebraic elements, minimal polynomials, vector spaces over a general field. [29,30]
• 11/30, 12/2: Algebraic extensions, finite extension, degree of towers of finite extensions, algebraically closed fields, "fundamental theorem of algebra" (statement only). [31]
• 12/4 (LAST CLASS): Proof that you can't trisect a 60 degree angle using the allowed moves of classical ruler-and-compass constructions. [32] (In the unlikely event that we get to this point early, we will do a bit more with fields.)
• 12/7, 12/9: Due to a last-minute change in the academic calendar, there is no class on these days, which are now part of a "Reading/Recitation/Review (RRR) period". I was planning to use the last one or two classes for review, but instead I will schedule an optional review section and hold some extra office hours during this period, at times TBA.
• 12/17: FINAL EXAM. Covers the entire on the course, with somewhat more emphasis on the later material.