Mathematics 305: Abstract Algebra I.
Instructor: Andres Caicedo.
Time: MWF 10:40-11:30 am.
Place: Education building, Room 221.
Office Hours: By appointment. See this page for details.
Text: Redfield, Robert H. Abstract Algebra. A concrete introduction. Addison Wesley, 2001. ISBN: 0-201-43721-X. If needed, I will provide references for additional topics not covered by the textbook.
Contents: The usual syllabus for this course lists
Introduction to abstract algebraic systems – their motivation, definitions, and basic properties. Primary emphasis is on group theory (permutation and cyclic groups, subgroups, homomorphism, quotient groups), followed by a brief survey of rings, integral domains, and fields.
The textbook by Redfield deviates some from this format, by presenting the subject in a somewhat historical fashion, with the goal of understanding what polynomial equations with integer coefficients can be solved. We will first introduce fields (focusing on number fields), then rings, and finally groups. Once all these elements are in place, we will cover the basics of Galois theory and possible additional topics, depending on time.
We will cover the book in order; our first goal is to cover its first three parts. I will be adding to (and correcting along the way) the syllabus on a week-by-week basis roughly detailing the content we expect to cover each week. Beyond the topics corresponding to Abstract Algebra per se, I will also focus on developing the students’ skill and understanding of abstract arguments, with particular care being paid to how they write and explain mathematical results.
The introduction of abstract algebra transformed the way mathematics is practiced and understood, so we will focus on both very concrete examples and very general (abstract) settings. The course may prove challenging but I expect it will also prove rewarding.
- January 20-23: Chapter 1.
- January 26-30: Chapters 1-2.
- February 2-6: Chapters 2-3.
- February 9-13: Chapters 3-5.
- February 17-20: Chapters 4-6.
- February 23-27: Chapters 6-7.
- March 2-6: Chapters 6-7.
- March 9-13: Chapters 7, 9, 13.
- March 16-20: Chapters 8, 9, 22 (only the part pertaining to equivalence relations).
- March 30-April 3: Chapters 8, 10, 11.
- April 6-10: Chapters 10-12. (The quotient ring construction to produce field extensions is not explicit in the book. Let me know whether you need written references.)
Prerequisites: 187 (Discrete and foundational mathematics) and 301 (Linear algebra).
Grading: There will not be exams. Instead, the grade will be determined based on homework.
I will frequently assign problems and provide deadlines. Some of these problems are routine, others are more challenging, a few may give you extra credit points due to their difficulty. Some of the problems will be computations, but expect that a significant amount are questions that require proofs. For some, collaboration is allowed, although each student should write their own solutions. If a group of students collaborate in a problem, they should indicate so at the beginning of their solutions. Also, if additional references are consulted, they should be listed as well. It may happen that while reading a different book you see a solution for a homework problem. This is fine, as long as it is not done intentionally, and I trust your honesty in this regard. For some problems, I will specify that no collaboration is allowed.
No problems will be accepted past their deadline, and deadlines are non-negotiable.
I will pay particular attention not only to the correctness of the arguments, but also to how the arguments are presented. Your final grade will be determined based on the total score you accumulate through the term.
It may be that you do not see how to completely solve a problem, but you see how to solve it, if you could prove an intermediate result. If so, indicate this clearly, as it may result in partial credit. On the other hand, the fact that you write something does not mean you will get partial credit.
If through the term if becomes clear that a particular topic requires special and immediate attention, I might decide to quiz on it. If this is the case, quizzes will be announced at least two lectures in advance, and the total percentage that quizzes add up to the score for the course will not exceed 20%.
Similarly, if it turns out that a final exam must be given, it will take the form of a take home test, which you may think of as a last homework set, with more stringent time limit requirements. The percentage that it will contribute to the score for the course will not exceed 20%.
Homework, quizzes, and the final exam may be based on additional topics not covered in the book.
Attendance to lecture is not required but highly recommended.
As the term progresses, I will be getting pickier on how you write your solutions. Introduce and describe all your notation. Use words as necessary; strings of equations and implications do not suffice. You may lose points even if you have found the correct answer to a problem but it is not written appropriately. Do not turn in your scratch work, I expect to see the final product. Be neat. I am not requiring that you typeset (or LaTeX) your solutions, but I expect to be able to read them without any difficulty. Additional remarks are encouraged; for example, if a problem asks you to prove a result and you find a proof of a stronger statement, this may result in additional extra credit points.
Once your total score is determined, I will then grade on a linear scale:
- If your final score is 90% or higher, you receive an A.
- If it is between 80 and 89%, you receive a B.
- If it is between 70 and 79%, you receive a C.
- If it is between 60 and 69% you receive a D.
- If it is 59% or lower, you receive an F.
- There may be a small curve up if the distribution of scores warrants this. Plus and minus grades might be used for grades near the top or bottom of a grade range.
I will use this website to post any additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.
- Homework 1, due February 6, at the beginning of lecture. [This homework was not graded.]
- Homework 2, due February 20, at the beginning of lecture.
- Homework 3, due February 25, at the beginning of lecture.
- Homework 4, due March 2, at the beginning of lecture.
- Homework 5, due March 11, at the beginning of lecture.
- Homework 6, due April 3, at the beginning of lecture.
- Homework 7, due April 10, at the beginning of lecture.