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Math 120: Groups and Rings
In Fall 2015 I taught Math 120 at Stanford University. The course assistant was Niccolò Ronchetti. For questions about the material and class discussions, we used the Math 120 Piazza page.
Material covered:
- Week 1: 1.1, 1.3, 1.4, 1.6 (9/21: 1.1; 9/23: 1.4, begin 1.6; 9/25: 1.6, 1.3, begin 1.7)
- Online assignment #1 (on Coursework, due 9/27): read 1.7, 2.1,
2.2 - Week 2: 1.7, 2.3, 2.4, 3.1 (9/28: 1.7, 9/30: 3.1, 10/2: 2.4, 2.3)
- Online assignment #2 (due 10/4): read 3.2 through Corollary 10, 3.3, 4.1, 4.2
- Week 3: 3.3, 3.2, 6.3, 4.3 (10/5: 3.2, 3.3, 10/7: 6.3, 10/9: start 4.3)
- Online assignment #3 (due 10/11): read 4.3 through Proposition 10, 4.2, statement of Sylow theorem in 4.5
- Week 4: 4.3, 4.2, 4.5
- Online assignment #4 (due 10/18): read 7.1 and 7.2 carefully, try to understand these examples of rings (OK to skip "Quadratic Integer Rings" on p229 and group rings on p236)
- Week 5: 7.1, 7.2, 7.3
- Online assignment #5 (due 10/25): 7.4, 7.6, skim 7.5 without getting overwhelmed by details
- Week 6: 7.4, 7.6, 8.3
- Online assignment #6 (due 11/1): 8.1 through Theorem 4, 8.2 through Corollary 8, 8.3 through Corollary 15
- Week 7: 8.3, 8.2, 8.1
- Online assignment #7 (due 11/8): 9.3, 10.1 through p339 (you can look at 340-341 if you want)
- Week 8: 9.3, modules, isomorphisms of modules, free modules and basis
- Online assignment #8 (due 11/15): §2.1–2.5 of Reid's "Undergraduate Commutative Algebra" (skip first two full paragraphs on p38)
- Week 9: structure theorem for f.g. modules over a PID; applications: linear algebra, classification of f.g. abelian groups
- Week 10: continue with modules over a PID: rational canonical form, torsion-free implies free.
- Last lecture: semisimple rings and representation theory of finite groups in characteristic 0 (number and dimension of simple representations).
WIM paper topics: 1) alternating group An is simple for n≥5, or 2) Banach–Tarski paradox. For alternating group: definition of An is 3.5, definition of simple group is on p102, simplicity of A5 is Theorem 4.3.12, simplicity for general n is 4.6 (there are other proofs, you can use another argument if you like). Source for Banach–Tarski: Brief exposition by Terry Tao, also sketch of the proof on Wikipedia.
The full syllabus for the course is available here.
Textbook: Dummit and Foote, Abstract Algebra (3rd ed), required.
Further resources:
- Statement from the Registrar concerning students with documented disabilities: "Students who may need an academic accommodation based on the impact of a disability must initiate the request with the Student Disability Resource Center (SDRC) located within the Office of Accessible Education (OAE). SDRC staff will evaluate the request with required documentation, recommend reasonable accommodations, and prepare an Accommodation Letter for faculty dated in the current quarter in which the request is being made. Students should contact the SDRC as soon as possible since timely notice is needed to coordinate accommodations. The OAE is located at 563 Salvatierra Walk (phone: 723-1066)."
- Stanford's Honor Code and Fundamental Standard.