**In Fall 2021, the mode of instruction for this class will be in-person.**
The fundamental topics of modern algebra including elementary number theory, groups, rings, polynomials and fields.
Prerequisites: A grade of C or better in MATH 2710 or 2142Q. Recommended preparation: MATH 2144Q or 2210Q. Cannot be taken after passing MATH 3231.
Meets: Tuesdays, and Thursdays, 2-3:15, at MONT 414.
About the Instructor: Álvaro Lozano-Robledo
Office hours: Mondays 11-12 and Fridays 2:30-3:30 (online, via Webex) or by appointment (in person at MONT 233).
About the Book and Other Resources
I will be following the main reference: Tom Judson’s “Abstract Algebra: Theory and Applications” (2020 Annual Edition) which you can find for free at the author’s website or you can purchase an inexpensive hard-copy. There are many other excellent references on abstract algebra, but I will only point out Prof. Keith Conrad’s website which contains many expository papers that you may find very useful.
During the course we will discuss computability aspects of the theory, so it is helpful for students to familiarize themselves with computer packages that can handle computations on elliptic curves:
- A list of videos intros to the LMFDB, Magma, and Sage/CoCalc.
- LMFDB: The L-functions and Modular Forms DataBase.
- MAGMA. There is an online calculator here.
- SageMath and CoCalc.
Homework sets will be due every other week. Most of what you learn in this course will be the result of working exercises that are designed to reinforce key concepts, develop skills, and test your understanding of the material. There will be textbook exercises due at the end of every other week on lecture material. Some of the exercises are straightforward, others are very complex. In general homework is due every other Thursday.
Late homework will not be accepted. Although it should be done daily, it will only be collected once every other week, on Thursdays, in my office (MONT 233) *before* 3:30PM. The first assignment is due on Thursday 9/9.
You are encouraged to talk with your classmates about the homework. If you have difficulties, do not waste time — get help! Please come to office hours!
All problems numbers below refer to the online version of the book.
- Homework Set 1 (due 9/9):
- Ch 1: 29.
- Ch 2: 2, 5, 15 (a and b), 16, 17
- Ch 3: 1 (b and f), 2, 4, 7, 10, 14, 17, 31, 41.
- Homework Set 2 (due 9/23):
- Ch 4: 1, 2 (you may use a calculator or computer!), 4 (b and d), 6 (note: D4 has 8 elements), 7 (the quaternions are defined in Example 3.15 in the book), 10, 22(only part (a) and definitely use a calculator/computer to proceed like in Section 4.3), 23, 26, 31, 34, 39.
- Ch 5: 1(b and d), 2(b, d, and f), 5.
- Homework Set 3 (due 10/7):
- Ch 5: 6, 8, 13, [17, 18, and 19 — all three are related], 27.
- Ch 6: 1, 2, 5(a and b), 7, 8 (suppose there is such an x value and raise it to the p-1), 9, 17, 21 (here phi is the Euler phi function).
- Homework Set 4 (due 10/21):
- Ch 9: 4, 5, 8, 10, 11, 12, 14, 18, 22, 24, 25, 34, 35, 41, 50.
- Homework Set 5 (due 10/4):
- Ch 10: 1(d), 4, 5, 7, 8, 9, 13, 14.
- Ch 11: 2(b and c), 3, 5, 7, 9, 12, 16.
Information about exams and grading for your class:
Your grade in the course will be determined by your performance on the two midterm exams, a final exam, and your lecture grade. The lecture grade consists primarily of homework and class participation. Your entire grade is out of 550 points (see below):
Here you will find information about midterms and exams.
- FIRST EXAM: Tuesday, September 28th (in-class)
- Covers Chapter 3, 4, and 5.
- Review for Exam 1
- SECOND EXAM: Thursday, November 11th (in-class)
- FINAL EXAM:
- Take-home exam with open book/notes policy.
- FIRST EXAM: Tuesday, September 28th (in-class)
The final exam will cover material from the entire course, but there will be an emphasis on material covered after the second prelim. No calculators are allowed on exams.
Grade: The grading will be based on Prelim 1 (100 points), Prelim 2 (125 points), the final exam (175 points) and a lecture grade (150 points). The lecture grade will be based on homework and class participation.
- If you cannot take an exam at the scheduled time, you MUST let your instructor know BEFORE the exam; you will almost certainly get an ’F’ on an exam if you miss it for any reason and then try to explain later.
- Incompletes will be given only under exceptional circumstances and then only to students who have a passing grade on a substantial part of the course. Do not expect to be granted an incomplete simply because you have fallen behind in the course.
|1||Preliminaries, and Chapter 3: Groups|
|2||Chapter 4: Cyclic Groups|
|3||Ch. 5: Permutation Groups|
|4||Ch. 6: Cosets and Lagrange’s Theorem|
|5||Ch. 6: Fermat’s and Euler’s Theorems (also EXAM 1)|
|6||Ch. 10: Conjugation, Factor Groups, Normal Subroups|
|7||Ch. 11: Group Homomorphisms and Ch. 9: Isomorphisms|
|8||Ch. 11: The Isomorphism Theorems and Ch. 9: Direct Products|
|9||Ch. 13: Finite Abelian Groups|
|10||Ch. 12: Matrix Groups and Symmetry|
|11||Ch. 12: Symmetry (also EXAM 2)|
|12||Ch. 14: Group Actions and the Class Equation|
|13||Ch. 15: Sylow Groups and Theorems|
|14||Ch. 15: Sylow Groups and Theorems|
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