Math 31: Topics in Algebra

Last updated August 17, 2012

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Course Information






The course will progress tentatively as follows. We will begin by reviewing the main ideas of sets and functions, and from there we will begin our study of group theory. This should take roughly the first half of the course, after which we will have the midterm exam. We will then spend two to three weeks on ring theory. The course will end with some special topics and possibly applications of algebra, depending on the interests of the class. A more detailed schedule will be available on the course webpage, and any updates to the course structure will appear there.

Since the reading and writing of proofs is high on our list of objectives, many aspects of the course will be designed to develop these skills. For example, you will be strongly encouraged to read the textbook regularly. Also, the first few x-hours will be used as ``workshops'' for developing proof-writing skills. These will be purely for your benefit, and if you've already taken a proof-intensive class, you need not come regularly. After the first few weeks, we may continue the workshops if there is need, or we may begin using the x-hours to cover extra material and/or special topics.

The following is a tentative daily schedule for the course. It will be updated throughout the term to reflect our actual progress through the material. The reading assignment listed for each day is meant to be read after that day's class. It will generally be comprised of the sections in the book that we intend to cover in the next lecture, and it should be read before coming to the next class. There is no need to understand every detail of the assigned reading, but seeing it beforehand will help. All reading assignments are from Herstein unless otherwise noted.

Since I sometimes deviate from the textbook, or cover topics that are not fully addressed in the book, I will be posting a PDF version of my lecture notes here. The plan is to update them on a weekly basis.

Lecture Notes (Updated 8/17/2012)


Brief Description
Reading Assignment
6/22 Overview: What is Abstract Algebra? Set Theory handout and/or Section 1.2, skim Section 1.5
6/23 Motivating examples of groups Section 2.1, Section 1.3
6/25 Definitions and examples of groups Sections 1.4 and 2.1
6/27 More examples; the symmetric and dihedral groups Section 2.2
6/28 (x-hour) Proof Workshop
6/29 The symmetric and dihedral groups, Basic properties of groups Section 2.3
7/2 Subgroups No reading. (Optional: Section 4 of Gallian)
7/4 NO CLASS - Independence Day
7/5 (x-hour) Proof Workshop
7/6 Cyclic Groups Section 2.4 (up to the statement of Lagrange's theorem)
7/9 Cyclic groups (continued); Equivalence relations Section 2.4 (beginning with Lagrange's theorem)
7/11 Cosets and Lagrange's Theorem Section 2.5 (through the corollary to Theorem 2.5.5)
7/12 (x-hour) Proof Workshop
7/13 Homomorphisms and isomorphisms Section 2.5 (beginning with normal subgroups)
7/16 Cayley's Theorem and Kernels of Homomorphisms Section 2.6
7/18 Kernels (cont.) and Quotient Groups Section 2.7
7/19 (x-hour)
7/20 Quotient Groups (cont.) and Normal Subgroups None
7/23 MIDTERM EXAM Section 2.7
7/25 Quotient Groups and the First Homomorphism Theorem Section 3.2 (Skim Section 3.1 if you want to review the basics of the symmetric group)
7/26 (x-hour)
7/27 The Symmetric Group: Cycle decomposition Section 3.3
7/30 Even and Odd Permutations; The Alternating Group Sections 2.9 and 2.10
8/1 Direct Products of Groups and The Fundamental Theorem of Finite Abelian Groups Section 2.10
8/2 (x-hour) The Fundamental Theorem of Finite Abelian Groups (cont.); the more general classification problem Sections 4.1 and 4.2
8/3 Rings Section 4.3
8/6 Ring Homomorphisms and Ideals Section 4.4
8/8 Quotient Rings and Maximal Ideals Section 4.5
8/9 (x-hour) Polynomial Rings Section 4.6
8/10 Irreducibility of Polynomials Sections 5.3, 5.4 and 5.6
8/13 Roots of Polynomials and Field Extensions Sections 5.3, 5.4 and 5.6
8/15 Field Extensions (continued) None
8/16 (x-hour) Presentations
8/17 Presentations; The Splitting Field of a Polynomial
8/20 Presentations
8/22 Recap and brief overview of Galois theory