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
In previous courses you have seen various kinds of algebra, from the algebra of real and complex numbers, to polynomials, functions, vectors, and matrices. Abstract algebra (mathematicians would just call this "algebra") encompasses all of this and much more. Roughly speaking, abstract algebra studies the structure of sets with operations on them satisfying some basic properties.
We will study three basic kinds of "sets with operations on them", called groups, rings, and fields because they have proven themselves to be the most useful algebraic structures, with the exception of abstract vector spaces. Most of you will recall their definitions from Math 311W. If not, please review them now.
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 set of integers with the operations of addition and multiplication. Another example is the ring of polynomials. We will not have time to investigate the ring of linear transformations on a vector space.
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. Fields that you have already run into include the real numbers and the rational numbers.
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. We will assume that you have developed some facility with proofs in Math 311W.
Some notes of Michael Hutchings giving a very basic introduction
to proofs are available here.
Suggestions
for writing mathematics.
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)
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.
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, list your collaborators on your homework. This will not affect your grade, unless it is clear that you have copied almost 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 justification, or even false. You can of course cite theorems that we have already proved in class or already covered from the book.
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. We will completely skip Part VIII on group theory in topology; this material is best learned in a topology course.
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.
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]