Time:11:10 AM – 12:15PM on Mondays and Wednesdays (lectures); 2:00PM – 3:15PM on Fridays (recitations)
Lecturer: Joel Spencer, spencer [at] cims [dot]nyu [dot] edu
Teaching Assistant: Mark Kim, markhkim [at] math [dot] nyu [dot] edu
Here are the student evaluations.
This is the second part of the two-semester sequence on abstract algebra at the advanced-undergraduate level, covering field theory. (The recitation page for the first part is available here.) The main textbook for the course is I. Herstein’s Topics in Algebra (2e). See Spencer’s algebra page for detailed information about the course, including the syllabus.
Here are some recommended supplementary references, with comments:
- E. Artin, Galois Theory. In the beginning, there was Évariste Galois, who laid the groundwork for two major subfields of abstract algebra: group theory and field theory. His major work dealt with what is now known as Galois theory, which is a study of algebraic extensions by exploiting various symmetry properties of the field operations. Then came Emil Artin, who transformed Galois theory to its current form: “Since my mathematical youth I have been under the spell of the classical theory of Galois. This charm has forced me to return to it again and again, and to try to find new ways to prove its fundamental theorems.” (Artin, 1950)\ Not only was Artin responsible for the modern formulation of Galois theory, he was also the single most influential expositor of the theory. It would not be an exaggeration to say that every account of Galois theory from the second half of the twentieth century onward is based on some incarnation of Artin’s lectures. Galois Theory is the distilled version of Artin’s exposition of Galois theory, starting at a basic enough level that a bright high school student could follow and yet covering all the essential elements of the theory at a brisk pace. Read it for the content, and read it again for the writing.
- M. Artin, Algebra (2e). One downside to Emil Artin’s little booklet is that its expository style diverges significantly from the modern textbook style of writing. To remedy this issue, one might take a look at the textbook by Michael Artin–the eldest son of Emil, and an expert algebraist in his own right. Chapter 15 (Fields) and Chapter 16 (Galois Theory) serve as a fantastic introduction to field theory, with plenty of exercises that are entirely absent in Emil Artin’s text. A note of caution: do not get the first edition; the second edition reads much, much better than the first.
- I. Stewart, Galois Theory (2e or 3e). This is The Standard Textbook that deals solely with Galois theory. Clear, thorough, and straight to the point. The third edition develops the theory over subfields of , which is what we will do in this course. Nevertheless, I should point out that people disagree on which edition of the text is better. The fourth edition is coming out in March, so it might actually be a good idea to check out that one instead of the third. If you do decide to go for the second edition or the third edition, be sure to refer to George Mark Bergman’s clarifications and corrections.
Office Hours will be held in WWH 1109 on Wednesdays from 3:30 pm to 4:30 pm. 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 (1/30) – P. Halmos, “How to Write Mathematics“; also, examples of great mathematical writing (MathOverflow)
- Recitation 2 (2/6) – See Sec. 6.1 of Ireland & Rosen, A Classical Introduction to Modern Number Theory for a quick introduction to algebraic integers and Sec. 13.1 – 13.3 of M. Artin, Algebra for an overview of algebraic quadratic integers with a view toward ideal class groups. Sec. 8.1 of Dummit & Foote, Abstract Algebra contains details on and .
- Recitation 3 (2/13) – See Sec. 3.1 of N. Jacobson, Basic Algebra, vol. 1 for the definition of modules via ring actions. Sec. 3.1 – 3.5 of Jacobson, as well as Sec. 14.1 – 14.3 of M. Artin, Algebra, cover the basic theory of modules that we went over in the recitation.
- Recitation 4 (2/20) – See Sec. 14.4 – 14.8 of M. Artin, Algebra and Sec. 3.6 – 3.10 of N. Jacobson, Basic Algebra, vol. 1 for a thorough introduction to the classification theorem on finitely generated modules over a principal ideal domain. The existence of a Smith normal form is easier to establish over Euclidean domains, as we need not appeal to the ascending chain condition on a Euclidean domain: substitute every ACC argument in the proof with a standard greatest-common-divisor argument, and we’re good to go.
- Recitation 5 (2/27) – See the MathOverflow page on different proofs of the fundamental theorem of algebra.
- 3/3 – March 3 is the birthday of Emil Artin, a towering figure in Galois theory and many other disciplines of mathematics. Here is a free version of Artin’s Galois Theory.
- Recitation 6 (3/6) – We reviewed a number of topics concerning the theory of field extensions in the first six sections of Chapter 15 of M. Artin, Algebra. The entire chapter is written extremely well, so give it a try if you have the time. See the Wikipedia article on formal derivatives, the Wikipedia article on the derivation-defintion of the tangent space of a manifold, and the Wikipedia article on Fréchet derivatives for more information on various generalizations of the notion of derivative. See also the Wikipedia article on centipede mathematics.
- Recitation 7 (3/13) – Matthew Grote led the recitation, surveying the history of abstract algebra.
- 3/20 – Spring break; no recitation!
- 3/23 – March 23 is the birthday of Emmy Noether, who gave us much of the ring theory and module theory we have studied in the course. Here is an English translation of Noether’s Idealtheorie in Ringbereichen.
- 3/25 – The midterm exam will take place in WWH 317 at 11am.
- Recitation 8 (3/27) – We talked about the historical motivation for Galois theory. See the Wikipedia page on the cubic formula and the wikipedia page on the quartic formula for the intuitive notion of solvabilty by radicals. See also the story of Tartaglia v. Cardano, a biography of Évariste Galois, and a biography of Emil Artin, all on MacTutor History of Mathematics.
- Recitation 9 (4/3) – Cyclotomic field on Wikipedia
- Recitation 10 (4/10) – Radical extensions on Wikipedia
- Recitation 11 (4/17) – How do we show that there are only three order-12 subgroups of ? Since a subgroup of is normal if and only if there exists a group and a surjective group homomorphism such that , it suffices to classify all surjective group homomorphisms on into . It is not hard to see that there are only three such mappings!
- Recitation 12 (4/24) – The inverse Galois problem is a classic open problem in abstract algebra. See this MathOverflow post for motivation, and this StackExchange post for a list of references. Do not confuse the inverse Galois problem with the problem of realizing finite groups as the Galois group of a finite normal extension of some field.
- Recitation 13 (5/1) – How to find the Galois group of a polynomial? on StackExchange
- Recitation 14 (5/8) – Where do we go from here? (Banach algebras in functional analysis and abstract harmonic analysis, Lie algebras and Lie groups in mathematical physics and differential geometry, applications of finite field theory to cryptography, etc., etc.)
- 5/15 – The review session will take place in WWH 312 at 5pm.
- 5/18 – The final exam will take place in WWH 317 at 10am.