Mathematics
351, Abstract Algebra
Fall 2010: Syllabus
Text: Hungerford, Abstract Algebra, an Introduction, 2nd edition
Instructor: Professor Robert Wilson rwilson@math
Office hours: Monday, 10:20 – 11:40 and Wednesday,
10:20 – 11:40 in Hill-340
or by appointment, in Hill 340
Home page: http://www.math.rutgers.edu/~rwilson
Telephone: 732-445-2390 Ext. 1317
Course web page: http://www.math.rutgers.edu/~rwilson/math351_fall_2010.html
In class (midterm)
examinations: Monday,
October 11; Monday, November 15
Final examination: Thursday, December 16, 8-11 AM. The location is TBA (but I would not bet against it being the regular classroom).
The course is about the general theory of algebraic operations. You have studied numbers, polynomials, functions, vectors and matrices and you have probably observed that computations with them have some broad similarities. This course goes into the theory common to these examples, a theory that has evolved at an ever-increasing rate over the last 100 to 200 years. One of our goals is to show how some much older problems (including these nice samples) can be solved using these more modern ideas. These simple ancient questions are pertinent to current problems such as the security of financial transactions on the internet (Hungerford, Chap. 12).
Most of the course will be devoted to the study of two types of algebraic objects: rings (which generalize the integers, i.e., they have operations of addition and multiplication) and groups (which generalize the set of all permutations of a set, i.e., there is a single operation generalizing composition). We will begin (Chapters 1 and 2) by recalling some properties of the integers. Then (Chapter 3 and part of Chapter 6) we will define rings and give some basic definitions and results about their structure. Next (Chapters 4 and 5) we will consider other important examples of rings, notably polynomial rings and then study some further structural properties of rings (in the remainder of Chapter 6). We will then begin the study of groups (Chapter 7). At the end of the course we will study some additional topics in group theory (Chapter 8) and ring theory (Chapter 9).
Here it the tentative schedule for the course
September 1 – Lecture on Sections 1.1, 1.2; Workshop #1
September 8 – Lecture on Sections 1.3, 2.1
September 13 – Lecture on Section 2.2, 2.3
September 15 – Lecture on Sections 3.1, 3.2.Workshop #2
September 20 – Lecture on Section 3.3
September 22 – Lecture on Section 6.1; Workshop #3
September 27 – Lecture on Section 6.2
September 29 – Lecture on Section 4.1; Workshop #4
October 4 – Lecture on Sections 4.2, 4.3
October 6 – Lecture on Section 4.4; Workshop #5
October 11 – Exam #1
October 13 – Sections 5.1, 5.2; Workshop #6
October 18 – Lecture on Section 5.3
October 20 – Lecture on Section 6.3; Workshop #7
October 25 – Lecture on Sections 7.1, 7.2
October 27 – Lecture on Section 7.9; Workshop #8
November 1 – Lecture on Sections 7.3, 7.4
November 3 – Lecture on Section 7.5; Workshop #9
November 8 – Lecture on Sections 7.6, 7.7
November 10 – Lecture on Section 7.8; Workshop #10
November 15 – Exam #2
November 17 – Lecture on Section 8.1; Workshop #11
November 22 – Lecture on Section 8.2; Workshop #12
November 29 – Lecture on Section 8.3
December 1 – Lecture on Section 8.4; Workshop #13
December 6 – Lecture on Section 9.1
December 8 – Lecture on Section 9.2; Workshop #14
December 13 – Lecture on Section 9.2
The material to be covered on exams will be announced in class, no later than two weeks in advance. Review materials will be posted in advance, and should be used together with workshop and homework assignments for preparation.
Course level:
This is a high-level course. You will be expected to understand the proofs
which are given in the text, and (especially) in lectures, and to construct
your own proofs. You should expect to become more experienced in this as the
term progresses. The course is one of two that satisfies the algebra
requirement for the mathematics major. The alternative is Mathematics 350 (advanced
linear algebra).
As a general rule, undergraduates should expect to spend approximately two hours outside of class for every hour spent in class. As Mathematics 351 is a 4-credit course, and is one of our more challenging courses,
Students in Mathematics 351 should be prepared to spend 8 to 10 hours per week on the course, in addition to the class meetings.
Writing proofs may be particularly time-consuming at first. If you get stuck
or just need feedback, paying a visit to office hours will probably be
worthwhile.
However: for this to be really useful, you should plan to discuss a
couple of specific problems that you have thought about carefully, and have
written as much as you can about. The more specifically you can describe your
line of thinking, and where you are stuck, the more productive the office hours
will be.
Calculator
A numerical calculator will certainly be useful (as
will become apparent during the first lecture). However, graphing is irrelevant. When
working with numerical data you will usually need to keep everything in exact
terms (21/2 rather than 1.414 for example), so the usefulness of the
calculator is limited to rather simple calculations. We do need to do some substantial
arithmetic occasionally, of the sort you would not want to do by hand. On
exams, a TI-83 will be permitted, but no calculator with alphabetic keyboard,
large memory or built-in algebra system will be allowed.
Grading:
|
Two in-class midterm examinations (20% each): |
40% |
|
Final examination: |
40% |
|
Workshops: |
10% |
|
Homework and Quizzes: |
10%: |
Homework and quizzes:
Regular homework will be assigned in class from the book. Quizzes will be given
based on the homework. They may be announced or unannounced. Questions
concerning the homework should be raised in class.
Workshops:
In the Workshop sessions you will work in groups on more difficult problems, to
be handed out in the workshop, under your professor's supervision. Some of
these problems, but not all, come from the book. Selected workshop problems
will be assigned to be written up and handed in.
Collaboration vs. Plagiarism1 :
The workshop involves a mixture of two very different kinds of work: collaborative
and independent. It is important to understand what this means in
practice.
- During the workshop session you are expected to collaborate fully with your group. The goal here is not only to solve the problems, but to develop a range of skills relating to technical communication and group work, and to reach a common understanding of the issues with your classmates.
- Outside the classroom you may also discuss these problems with your classmates, but you should mention explicitly any such collaboration in your write-up. Unacknowledged collaboration is considered plagiarism, which is a serious violation of the principles of academic honesty, and which the professor takes seriously. If two students' write-ups are identical, that is very strong evidence of plagiarism. The line between legitimate collaboration and plagiarism is subtle, but you can steer clear of it by making it a habit to acknowledge both your collaboration with others and your use of outside sources such as books and websites. Then you can make maximum use of collaboration.
- The final write-up is your
own.
This means that it is entirely written in your own words, even if you like someone else's words better, and even if a classmate has come up with the key idea!
The only person whom you are permitted to consult about the write-up is the professor. Consult the professor if you have trouble with the presentation, or whenever the rules concerning write-ups are unclear in practice.
Main
page, 351.
1Acknowledgement: Substantial portions of this page are adapted or copied from Professors Sims', Lyon's and Cherlin’s 351 pages from the period 2001-2006.