Math 113: Abstract algebra

Instructor

Michael Hutchings
hutching@math.berkeley.edu
Office phone: 510-642-4329.
Office: 923 Evans.
Office hours: Tuesday 9:30 - 11:30, 1:00 - 2:00 (subject to change or temporary rescheduling), or by appointment.
Office hours the week of 5/19 to 5/23: Tuesday, Thursday, and Friday, 3:00 to 5:00 PM.

In addition to my office hours, Eli Grigsby will hold office hours for all Math 113 students, in room 891 Evans, on Wednesdays from 10-3pm and Thursdays from 8-11am and 4-6pm.

Spacetime coordinates

The course meets in 70 Evans on Tuesdays and Thursdays at 8:00 AM.

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 encompasses all of this and much more, and delves deeply into questions of ``what is going on'' with algebra in general. 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 form a very interesting 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 develop 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.

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, 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 abstract algebra texts out there, and if you want some additional perspectives you might try browsing through these in the library or your favorite bookstore.

Equipment

It is recommended that you obtain a 3x3 Rubik's cube, if you do not already have access to one, e.g. from rubiks.com. This will be used to illustrate some group-theoretic concepts in a fun, hands-on way.

Rubik's cube has been extensively studied by cube enthusiasts over the last 25 years, and I recently discovered this webpage which has links to enormous amounts of amazing information about it. For example, section 6 of that webpage includes links to computer programs which in a matter of days can solve a scrambled cube with the absolute minimum number of turns, and which within seconds can come within a couple of turns of the minimum! As far as I know the minimum is always at most 22 or 24 turns (depending on whether you count a 180 degree turn of a face as one turn or two), and usually a bit less, much shorter than human algorithms for solving the cube which may require 60 turns or more. There is a beautiful trick using a little group theory (namely cosets, which we will learn about) which allows the computer to quickly find a nearly minimal solution, and also to dramatically prune the search tree in order to find a minimal solution in a reasonable amount of time.

Notes on how to write proofs

I put together some old notes which give a very basic introduction to mathematical proofs. Here they are: pdf postscript.

Approximate syllabus

The following is a rough, tentative plan for the topics we will cover. What we actually end up doing in class will be listed below after each lecture.
• Preliminaries. We will begin with a brief review of some preliminary topics, including sets, relations, induction, and some very basic number theory. Some of this material is in section 0 of the book, some is scattered throughout random later sections, and some is not in the book at all. (Some is covered in the notes 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 do some additional group theory from Part VII as time permits.
• 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 Part 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, and maybe a bit of Part X. We will develop enough machinery to prove that one cannot trisect a sixty degree angle with a ruler and compass.

What we actually did in class

• 1/21: About the course; review of sets and functions. (Some of this is in Section 0 of Fraleigh.)
• 1/23: Equivalence relations and modular arithmetic (see Section 0). Review of proof by induction (see the above notes).
• 1/28: Strong induction and existence of prime factorizations. The well-ordering principle and the division theorem (in the notes). Uniqueness of prime factorizations.
• 1/30: Conclusion of preliminaries: greatest common divisor, euclidean algorithm, and solving ax = b (mod n). Introduction to binary operations (see sections 1, 2, and 3 of Fraleigh).
• 2/4: Isomorphism of binary structures (see section 3).
• 2/6: Groups (see section 4). Introduction to nontrivial examples including Z_n^*, the Rubik's cube group (regarded as a subgroup of S_48), and dihedral groups.
• 2/11: More examples of groups including O(n), SO(n), the Klein 4-group, and symmetry groups of polyhedra. Subgroups (see section 5). Examples including subgroups of D_2, Z_4, and D_3, and some of the many subgroups of the Rubik's cube group.
• 2/13: The subgroup of a group generated by a subset of the group; examples. Cyclic groups. Every cylic group is isomorphic to Z or Z_n. Subgroups of cyclic groups are cyclic. The subgroup of Z generated by a and b is generated by gcd(a,b). (See section 6 and some of section 7. We're not doing Cayley graphs, but you can read about them for fun.)
• 2/18: Permutations in detail. Every permutation is a product of disjoint cycles (uniquely up to reordering of the factors). Every permutation is a product of transpositions. Definition of even and odd permutations. (See section 9 and some of section 8.)
• 2/20: Proof that no permutation is both even and odd. The alternating group. Statement of Lagrange's theorem and proof of basic corollaries. (See section 9 and some of section 10.)
• 2/25: Review for the midterm. Sam Lloyd's "15" puzzle. Cosets (see section 10).
• 2/27: We had the first midterm. Here are solutions. Here is the score distribution.
• 3/4: Proof of Lagrange's theorem (section 10). Direct products (section 11). Statement of the fundamental theorem of finitely generated abelian groups. (A proof, which is elementary but complicated, is given in section 38.) By the way, the book I recommended in class is Godel, Escher, Bach by Douglas R. Hofstadter.
• 3/6: Homomorphisms (section 13). Normal subgroups. The quotient of a group by a normal subgroup (section 14). (Quotient groups are called "factor groups" in the book.)
• 3/11: More about normal subgroups and quotient groups. The "fundamental homomorphism theorem". Examples. (See sections 14 and 15.)
• 3/13: Commutator subgroup and abelianization. Converse of Lagrange's theorem is false (section 15). But Cauchy's theorem provides a partial converse. Statement of the Sylow theorems (section 36). (We don't have time for more advanced group theory in this course because we need to do the basics of rings and fields. But you might enjoy reading sections 36 and 37.)
• 3/18: Rings. Lots of definitions and basic examples. (See section 18.)
• 3/20: Integral domains. The field of quotients of an integral domain. (See sections 19 and 21.) (You might also want to read section 20 for fun; we did some but not all of this material before.)
• 3/25 and 3/27: Nothing (spring break).
• 4/1: The ring of polynomials R[x]. If R is an integral domain then so is R[x]. A polynomial determines a function R -> R, but the function does not always determine the polynomial. Evaluation homomorphisms R[x] -> R (need R commutative). Division theorem for F[x] where F is a field. The factor theorem. Algebraically closed fields. (See section 22 and some of 23.)
• 4/3: Irreducible polynomials. Over a field, a polynomial of degree 2 or 3 is irreducible iff it has a root. If f in Z[x] is irreducible over Z then it is irreducible over Q. (The book doesn't give the direct proof that we gave, although an elegant more general statement is proved in section 45.) Corollary: rational root theorem. The Eisenstein criterion (see section 23).
• 4/8: If p is a prime number then Z_p^* is cyclic (see section 23). If F is a field of characteristic zero then there is a unique nonzero homomorphism from Q to F. Review for the midterm.
• 4/10: We had the second midterm. Here are the questions with solutions. Here is the score distribution.
• 4/15: Greatest common divisor and Euclidean algorithm in F[x]. (The book doesn't really do this, although there is a generalization in section 46.) Unique factorization in F[x], using the above. (For a different proof see sections 23 and 27.) Z[sqrt{-5}] is not a unique factorization domain. (See Example 47.9 and the definitions at the beginning of section 45.)
• 4/17: Ideals and quotient rings (section 26). Started on prime and maximal ideals (section 27).
• 4/22: We covered the rest of the material in section 27, but I tried to organize it a little more clearly than in the book. Next we will be doing Part VI (Extension fields), which will be the climax of the course.
• 4/24: Field extensions, algebraic numbers, minimal polynomial, etc; see section 29.
• 4/29: More about simple algebraic extensions, see section 29. Review of vector spaces over an arbitrary field, see section 30.
• 5/1: Algebraic extensions, finite extensions, degree of towers of finite extensions (see section 31).
• 5/6: We proved that you can't trisect a 60 degree angle with a ruler and compass, cf. section 32.
• 5/8: We proved the classification of finite fields, see section 33. This is a long and not so easy proof, but it brings together a lot of the concepts we have been studying lately so it is a good review.
• NOTE: on 5/13 there will be a guest lecture by Prof. Tara Holm. Although new material in this lecture will not be on the final, the lecture will bring together more of the previous course material, so it is strongly recommended for review. I will be out of town for the week of 5/12 to 5/16, but I will hold office hours the following week, on Tuesday, Thursday, and Friday from 3:00 to 5:00 PM.

Homework assignments

• HW#1, due 1/30. pdf ps Selected solutions. pdf ps
• HW#2, due 2/6. pdf ps Selected solutions. pdf ps
• HW#3, due 2/13. pdf ps Selected solutions. pdf ps
• HW#4, due 2/20. pdf ps Solutions. pdf ps
• No homework due 2/27 because of the first midterm. The first midterm will cover what we do through 2/20.
• HW#5, due 3/6. pdf ps Selected solutions. pdf ps
• HW#6, due 3/13. pdf ps. Sorry, no solutions this week.
• HW#7, due 3/20. pdf ps Selected solutions. pdf ps
• No homework due 3/27 because of spring break.
• HW#8, due 4/3. pdf ps. Solutions. pdf ps
• No homework due 4/10 because of the second midterm. The second midterm will cover what we did through 4/3 (except that the Sylow theorems will not be covered).
• HW#9, due 4/17. pdf ps. Solutions. pdf ps
• HW#10, due 4/24. Section 26, problems 3, 10, 15, 16c, 22, 30, 31. Section 27, problems 6, 14, 15, 16. Section 45, problem 10. Extra credit: section 27, problem 38. Solutions. pdf ps
• HW#11, due 5/1. Section 27, problem 32. Section 29, problems 4, 6, 8, 10, 12, 14, 16, 23, 26, 29, 30, 36, 37. Solutions. pdf ps
• HW#12, due 5/8. Section 30, problems 10, 15, 26. Section 31, problems 8, 10, 12, 19, 23, 24, 25, 30, 31, 34, 36. Solutions. pdf ps
• Final exam review.

Homework policy

Homework will be due on Thursdays at 8:10 AM 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!) Assignments will be posted below on the preceding Thursday night. 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. 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, and 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 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." 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, and resist the temptation to use phrases such as "it is easy to see that...". You can of course cite theorems that we have already proved in class or from the book.