This course is a continuation of last semester's Math 311. We will cover topics in group theory, including the Sylow theorems, abstract vector spaces and modules, and field theory, including Galois theory.
Prerequisite: Math 311
Time: MWF 9 AM
Lecturer: Rebecca Lehman
Office: Gibson 412. Office Hours: Mon 1-3 PM, Tues 10-12 AM; other times by appointment
Required text: Artin, Algebra, Prentice Hall 1991
Problem Sets: 30%
Midterm: 15%
Final: 30%
Presentations: 25%
My main goal in this class is to help you improve your skills in reading, constructing and explaining proofs. This class will be significantly more conceptual and proof-based, and less computational than last semester. You will notice that I have devalued the exams. The purpose of the exams is to spot check your knowledge of the definitions, and to give you a few clever little problems to think about.
I will collect a few homework problems every week instead of periodic large problem sets; I hope that this will enable you to pay more attention to the details of your writing, and me to write more detailed comments.
You will each also have the opportunity to give two oral presentations on applications of the theory. Unlike last year, the topics will be chosen from the textbook and will not require much outside research on your part; I want you to concentrate on making your exposition clear, complete and concise. I will meet with you to help you plan and practice before your talk.
Don't Panic! Collaboration is encouraged, with your colleagues, with me and with anyone else you can find to collaborate with.
All the problems in Artin are good, and all the unstarred problems should be manageable. I will post one or two problems here for each lecture. Problems will be due one week from the Monday after the lecture is given.
Group Theory: Jan. 14-Feb. 8.
Mon, Jan 14, Introduction. Review 311 topics, Artin Chapters 1-2, 5.
Wed, Jan 16: Read Section 6.1. Homework problem (Due Jan 28): 6.1.6.
Fri, Jan 18: Read Section 6.3 (and 6.2 if your Algebra I didn't include the platonic solids). Fill in the gaps in proving the classification of groups of order 8. Think about the ways these groups could act on a set of order 6. Homework problem (due Jan. 28): 6.3.8.
Mon, Jan 21: MLK Day--No Class
Wed, Jan 23: Read Section 6.4. Homework problem (Due Feb 6): 6.4.9.
Fri, Jan 25: Section 6.4: Examples. Try at least a couple of the following problems: 6.3.15-17, 6.4.2, 6.4.3, 6.4.6, 6.4.15. Homework problem (Due Feb. 6): 6.4.16.
Mon, Jan 28: Read Section 6.7. Homework problem (Due Feb. 11): 6.7.3.
Wed, Jan 30: Read Section 6.8. Homework problem (Due Feb. 11): 6.8.13.
Fri, Feb. 1: Homework problem (Due Feb. 11): 6 Misc. #2
Mon, Feb. 4: Mardi Gras Eve--No Class
Presentation:
Classifying the Groups of order 12--Travis, Feb 8.
Vector Spaces and Modules: Feb. 6-Feb. 22.
Wed, Feb. 6: Read Section 12.1. Homework problem (Due Feb. 18): 12.1.7.
Mon, Feb. 11: Finish Section 12.1. Homework problem (Due Feb. 25): 12.1.12.
Wed, Feb. 13: Read Section 12.2. Homework problems (Due Feb. 25): 12.2.3, 12.2.6 (Hint: these two problems are related.)
Fri, Feb. 15: Read Sections 12.3-12.4. Homework problem (Due Feb. 25): 12.3.3.
Mon, Feb. 18: Finish Section 12.4. Homework problem (Due Mar. 3): 12.4.7.
Wed, Feb. 20: Read Section 12.5. Homework problem (Due Mar. 3): 12.5.8.
Fri, Feb. 22: Homework problem (Due Mar. 3): 12.5.4.
Presentations:
The Structure Theorem for Abelian Groups--Eddie, Mar.7
Modules of Linear Operators--Jacek, Mar. 10
Free Modules over Common Rings--Chris, Mar. 12
Midterm Exam: March 5
Here is a list of topics for the midterm: Midterm Review Topics. We will discuss problems on Monday.
Here are the solutions to the midterm exam: Midterm Exam. Please ask if you have any questions on them. With the exception of the Cayley-Hamilton Theorem (which perhaps I should not have assumed you knew) everything on this midterm is fair game for the final.
Fields and Galois Theory:
Mon, Feb. 25: Read Sections 13.1-13.2. Homework problem (Due Mar. 10): 13.1.3.
Wed, Feb. 27: Read section 13.3. Homework problem (Due Mar. 10): 13.3.11.
Fri, Feb. 29: Read Section 13.5-6. Homework problem (Due Mar. 10): 13.5.1.
Fri, Mar. 15: Finish Section 13.6. Homework problems (Due Apr. 7): 13.6.8, 13.6.11
Wed, Mar. 26: Read Sections 13.8-13.9. Homework problems (Due Apr. 7): 13.8.3, 13.9.2
Mon, Mar. 31: Read Section 14.1. Homework problems (Due Apr. 14): 14.1.8, 14.1.15.
Wed, Apr. 2: Finish Section 14.1. Homework problems (Due Apr. 14): 14.1.17, 14.3.4.
Mon, Apr. 7: Read Sections 14.3-14.4. Homework problems (Due Apr. 21): 14.3.5, 14.4.2.
Wed, Apr. 9: Finish Section 14.4. Homework problems (Due Apr. 21): 14.4.1, 14.5.1.
Mon, Apr. 14: Read Section 14.5. Homework problems (Due Apr. 28): 14.5.4, 14.5.8.
Wed, Apr. 16: Read Section 14.7. Homework problem (Due Apr. 28): 14.7.1, 14.7.3. What about general pth powers?
Fri, Apr. 18: Problem session. Look at 14.1: 2,3,7,12,19; 14.4: 2-3; 14.5: 3,7,9,11. Bring your questions; otherwrise I'll pick the examples.
Mon, Apr. 21: Read Section 14.8.
Fri, Apr. 25: Read Section 14.9.
Mon, Apr. 28: Here is a list of review topics and suggested problems for the final. Final Review Topics. We will discuss some of the problems on Monday. Focus on the finite fields and the Galois theory problems. Please hand in your last problem set on time so that I can grade it.
Presentations:
Ruler and Compass Constructions--Eddie, Mar. 28
Function Fields and Riemann Surfaces--Chris, Apr. 4
Galois Theory of Cubic Equations--Jacek, Apr. 11
Galois Theory of Quartic Equations--Travis, Apr. 23
Final Exam: Here are the solutions to the final exam: Thank you for taking Algebra, and enjoy your summer!
Here is an old solution set of mine for use as a template: Sample Problem Set. If you're just starting out, ignore all the headers above "\begin{document}," then insert your own problems and solutions into the template.
This "Simplified Introduction" is a helpful reference guide: Simplified Introduction to LaTeX.
More LaTeX resources are here: MIT Math Resources.
LaTeX Resources:
The use of TeX/LaTeX is not required in this class, but is recommended, especially for those who expect to continue in math. Here are some resources for those who wish to try. You can also ask me or anyone else for help.