Time:11:10 AM – 12:15PM on Tuesdays and Thursdays (lectures); 2:00PM – 3:15PM on Fridays (recitations)
Lecturer: Dmitry Zakharov, dz31 [at] nyu [dot] edu
Teaching Assistant: Mark Kim, markhkim [at] math [dot] nyu [dot] edu
Here are the student evaluations.
This is the first part of the two-semester sequence on abstract algebra at the advanced-undergraduate level, covering group theory and ring theory. The main textbook for the course is I. Herstein’s Topics in Algebra (2e). See the NYU Classes page (limited access) for the syllabus. Here are some recommended supplementary references, with comments:
- M. Artin, Algebra (2e). This is a decent, but not ideal, embodiment of what I believe is the best honors undergraduate abstract algebra curriculum in the country. The flow of the presentation is beyond excellent and allows for a good amount of advanced material to be covered with relative ease. Nevertheless, the details are rather sparse, and quite a few proofs in the book are nothing more than intuitive sketches. The high points of the book are Chapter 6 (Symmetry) and Chapter 9 (Linear Groups), which contain an excellent exposition of the more geometric aspects of group theory. Chapter 15 (Fields) and Chapter 16 (Galois Theory) are also fantastic–after all, Michael Artin is the son of Emil Artin, the great expositor of Galois theory–but these are less relevant to this course.
- D. S. Dummit and R. M. Foote, Abstract Algebra (3e). Every page of this book is eminently readable, but I have yet to meet a single person who has read the entire book. It is an excellent reference, to be sure: it has all the details you would want and more. Indeed, isn’t it comforting to know that Chapter 5 (Direct and Semidirect Products and Abelian Groups) and Chapter 6 (Further Topics in Group Theory) most likely contain all of the examples of groups you would encounter in this course? Plus, the organization of the book makes it easy to jump around without having to read sequentially. The chapters on ring theory here are also more detailed (but perhaps less fun to read) than those in Herstein’s book or Artin’s book.
- J. J. Rotman, An Introduction to the Theory of Groups (4e). This book is pitched at a slightly more advanced level than Herstein, Artin, or Dummit & Foote. Nevertheless, it is worth checking out for Chapter 1 (Groups and Homomorphisms) alone, which contains a nice semi-historical introduction to group theory via permutation groups. The proofs in this book are generally cleaner and shorter than those in the other books mentioned above, but this isn’t necessarily a good thing for an introductory course.
- L. C. Grove and C. T. Benson, Finite Reflection Groups (2e). Of relevance to this course is Chapter 2 (Finite Groups in Two and Three Dimensions), which details the theory of two-dimensional and three-dimensional symmetries in the group-theoretic context. The rest of the book is good, too, but has nothing to do with this course.
- M. Reid, Undergraduate Commutative Algebra. Chapter 1 (Basics) and the beginning sections of Chapter 6 (Rings of Fractions and Localisation) present a swift overview of a bulk of the ring-theoretic material that will be covered in this course. This book shouldn’t be your first reference, but it’s a nice one to skim through before the final exam to make sure you’ve understood the material. Section 9.9 (The Problem of Algebra in Teaching) is also a fun read.
Office Hours will be held in WWH 1109 on Mondays, from 1:30PM to 3PM. You may also email me to ask questions or schedule an appointment.
It is strongly recommended that you typeset your homework write-ups with LaTeX (pronounced “lay-tec” or “lah-tec”), which is a document preparation system developed by Leslie Lamport as an extension of Donald Knuth‘s TeX typesetting system. With LaTeX, diagrams like
can be typed with relative ease. Besides, all professional documents in mathematics are written in LaTeX, so taking some time now to learn LaTeX is certainly a worthwhile investment.
In order to use LaTeX (comfortably), two things must be installed: a LaTeX engine and an integrated development environment for LaTeX. The instruction depends on your operating system:
- Windows – Go to the MiKTeX project page and download the current version of MiKTeX. I would recommend downloading the Net Installer, but you can go for the Basic Installer if you don’t have a lot of disk space on your computer. The installation process should be self-explanatory—should you require extra assistance, the MiKTeX download page comes with a user-friendly manual. Once the installation process is over, fire up TeXworks, which should have already been installed along with your MiKTeX distribution. If you don’t like TeXworks, then try TeXnicCenter. If you are used to the famous Java IDE Eclipse, then you might want to try out TeXlipse instead.
- Mac – Go to the MacTeX page and download the current version of MacTeX. The full MacTeX package is recommended, but you can donwload the BasicTeX package and the corresponding MacTeX-Additions if you don’t have a lot of disk space on your computer. In either case, the installation comes with two editors—TeXShop and TeXworks. Alternatively, you could install Latexian. TeXlipse works on Mac as well.
- Linux – Install TeXLive, which can be done through most major distros’ package managers. Download TeXLive-Full if you can, but do go for the basic version if disk space is at a premium. As for an IDE, I would recommend Kile, which is what I use. If you don’t like KDE software, then you will have to find your own! (Of course, you can use gedit, emacs, or vim, but let’s not go there.)
Once the installation process is over, download this sample file. Open it with your editor of choice and play around with it for a bit! (Not sure how the output file should look like? Here’s the pdf file.)
- Recitation 1 (9/12) » Naïve Set
» Complete classification of groups of small order: “List of small groups” on Wikipedia
» Algebraic reasoning vs. geometric reasoning vs. analytic reasoning: Introduction to The Princeton Companion to Mathematics (NYU NetID required)
» On the usefulness of abstract mathematics in mathematics education, and other topics: H. Wu, “The Mis-Education of Mathematics Teachers“
- Recitation 2 (9/19) » How to Read Commutative Diagrams
- Recitation 3 (9/26) » Constructing Homomorphisms via Structure-Preserving Extensions
- Recitation 4 (10/3) » Some examples of Group
» See Chapter 2 of Finite Reflection Groups by Grove and Benson (NYU NetID required) for details on finite symmetry groups in two and three dimensions.
» Alternatively, Chapter 6 of Artin’s Algebra also contains similar material, written in a more beginner-friendly style; the book is on reserve at the Courant Library.
- Recitation 5 (10/10) » “Fundamental
groups” on Wikipedia
» “Particle physics and representation theory” on Wikipedia
» “Group theory and its application to chemistry” on Chemwiki
- Recitation 6 (10/17) » The review session was held on Friday the 17th at 4:30pm.
- Recitation 7 (10/24) » Direct products and semidirect products, from Semidirect Products and Extensions
- *** Lecture on 10/28 » The Sylow theorems, from Group Actions
- Recitaiton 8 (10/31) » Fields
- Recitation 9 (10/7) » The Euclidean algorithm, , and
- *** Lecture on 11/13 » The Ring of Integers
- Recitation 10 (11/14) » The Ring of Integers, continued; and .
- Recitation 11 (11/21) » We classified all prime elements of