Math 113: Abstract algebra

Instructor

Graduate student instructor

Spacetime coordinates

Course goals

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

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

More about Rubik's cube

Notes on how to write proofs

Approximate syllabus

• 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

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, 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.

Exams and grading

Make-up exams will be allowed only in extreme circumstances, and may be in a different format such as an oral exam.

Please note that exam grades cannot be changed 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, and regrading one student's exam would not be fair to everyone else.

The course grade will be determined as follows: homework 30%, midterms 15% each, final 40%.