Math 1530: Abstract algebra, Spring 2017
TTh 1-2:20 Biomed 291

Instructor: Melody Chan
office: 311 Kassar House
course email: (please use this email address for course-related emails)
course website:
office hours, Tuesdays 11:30-12:30, Thursdays 4-5

Teaching Assistant: Michael Zellinger
email: michael_zellinger at brown dot edu
TA Office hour: 4-6pm Mondays, Kassar House Foxboro Auditorium

Course description

This is a semester course in abstract algebra, covering groups, rings, and fields. It also serves as a rigorous introduction to mathematical thinking and proofs. The textbook is Dummit and Foote's Abstract algebra, third edition. This book is often also used in Math 1540, the sequel to Math 1530 offered every spring. We will cover much of Chapters 1-5, 7-9, and 13.

The prerequisite for this class is Math 0520 or Math 0540 (linear algebra). If you do not have these prerequisites, are willing to work hard, and wish to take Math 1530 anyway, please talk to me. I will expect a good amount of work outside class on the problem sets. If you are unsure about whether to take this course, please feel free to discuss it with me.


Groups: definition and examples; symmetric, dihedral, matrix groups. Subgroups, quotients, simple groups. Group actions, Sylow theorems, direct and semidirect products. Rings: definitions and examples; homomorphisms, ideals, quotients. Polynomials, irreducibility and factorization, principal ideal domains and Euclidean domains. Fields: finite fields, finite and algebraic field extensions.

Required textbook

Dummit and Foote, Abstract Algebra, third edition.
This book is available at the bookstore for $135.33 new, $103 used.


Homework 20%, midterm 30%, final 50%.


Homework will be assigned and due every Tuesday. It must be handed in at the beginning of class in hard copy and stapled. No late homework will be accepted. However, your lowest homework score will be dropped.

Problem Set 1, due Tuesday January 31
Problem Set 2, due Tuesday February 7
Problem Set 3, due Tuesday February 14 accepted Thursday February 16 because of snow
Problem Set 4, due Tuesday February 28
Problem Set 5, due Tuesday March 7
Problem Set 6, due Tuesday March 14
Problem Set 7, due Tuesday March 21 due Thursday March 23
Problem Set 8, due Tuesday April 4
Problem Set 9, due Tuesday April 11
Problem Set 10, due Tuesday April 18
Problem Set 11, due Tuesday April 25
Problem Set 12, due Tuesday May 2

Solutions to selected problems

Exams and schedule

The midterm will be held Wednesday March 15 in the evening, 6-8pm, in Barus and Holley 166. If you have a legitimate conflict with this time, you must let me know by February 15. The final exam will be held Thursday May 11, 2-5pm in Barus and Holley 168.

Reading week schedule: There will be class as usual on Tuesday, May 2 and Thursday, May 4. There will be an optional office hour on Tuesday, May 9.

Class Date Topics Reading
1 Thurs Jan 26 Sets, functions, binary operations. Definition of a group p.1, p.15-16, this article
2 Tues Jan 31 Equivalence relations. The group Z/nZ p. 3, p. 8-9
3 Thurs Feb 2 (Z/nZ)* and arithmetic in Z/nZ. Some properties of groups. Order p. 10, p. 18-20
4 Tues Feb 7 Injective, surjective, bijective functions. Dihedral groups, symmetric groups. p. 2, pp. 23-25
5 Thurs Feb 9 Symmetric groups. Homomorphisms, isomorphisms. (Video lecture due to snow) pp. 29-32, pp. 36-39
6 Tues Feb 14 Homomorphisms, subgroups, generators. parts of p. 36, 46-48, 50, 61-63
7 Thurs Feb 16 Greatest common divisor and least common multiple; Euclidean algorithm. Cyclic groups pp. 4-5, 54-56
8 Thurs Feb 23 Group actions. Examples, permutation representation. (Guest lecture, Michael Zellinger) pp. 41-44, 112-114
9 Tues Feb 28 Cyclic groups again, lattice of subgroups. Image and kernel. Cosets. pp. 57-59, 66-67,
10 Thurs March 2 Cosets, normal subgroups, conjugation and conjugacy classes. pp. 77, 82, 89, 123
11 Tues March 7 Normal subgroups again; kernels again. Definition of the quotient group. First isomorphism theorem pp. 81-83, 97
12 Thurs March 9 The isomorphism theorems. pp. 98-100
13 Tues March 14 Snow day
14 Thurs March 16 Simple groups. Composition series and the Jordan Holder theorem. Signs of permutations; the alternating group. pp. 97, 101-103, 106-110
15 Tues March 21 Midterm pep talk. Direct products; the fundamental theorem of finitely generated abelian groups. Semidirect products. pp. 152-155, 158-159, 171, 175-176.
16 Thurs March 23 Semidirect products, Sylow theorems (statement).
17 Tues April 4 Rings, basic properties. Examples: polynomial rings, matrix rings, group rings. Definition of integral domain and field. 7.1, 7.2
18 Thurs April 6 Homomorphisms, ideals, and quotients of rings. 7.3
19 Tues April 11 Ideals generated by elements. Principal ideals; ideals of Z. Maximal ideals. (Guest lecture Michael Zellinger) 7.4
20 Thurs April 13 Maximal ideals and prime ideals. Sums, products, intersections of ideals. end of 7.3, 7.4
21 Tues April 18 Norms, Euclidean domains, division algorithm. Euclidean domains are principal ideal domains. 8.1
22 Thurs April 20 Principal ideal domains, unique factorization domains. The field of fractions of an integral domain. 8.2, 8.3
23 Tues April 25 PIDs are UFDs. Polynomial rings over UFDs are UFDs. 8.3, 9.3
24 Thurs April 27 Fields. Characteristic and prime subfield. Field extensions and degree. Adjoining a root of a polynomial. 13.1
25 Tues May 2 Algebraic field extensions. 13.1, 13.2
26 Thurs May 4 Algebraic field extensions. Impossibility of trisecting angles etc. 13.2, 13.3


You are welcome to collaborate with other students in the class on your homework, although I suggest that you think carefully about each problem on your own first. You are required to write up your solutions separately and write the names of the students with whom you worked on the assignment. (You may only use the Internet as a general reference, at the level of generality of Wikipedia.)

How much time will this class take?

Roughly speaking, you should spend ten hours every week outside of class, including attending Michael's problem session, my office hours, reviewing class material and doing homework. Attending the TA problem session is strongly encouraged. In addition to three hours of class every week and about twenty hours of additional exam preparation, I estimate a total of 189 hours of time spent on this class (39 class hours, 130 studying hours and 20 exam studying hours).

Accommodations for students with disabilities

Any student with a documented disability is welcome to contact me as early in the semester as possible so that we may arrange reasonable accommodations. As part of this process, please be in touch with Student and Employee Accessibility Services by calling 401-863-9588 or online at