Fall 2007: Syllabus

**Text:** Hungerford, *Abstract Algebra, an Introduction*, 2^{nd}
edition

**Instructor:** Professor Robert Wilson rwilson@math

*Office hours:* Monday, 10:20 – 11:40 and Tuesday,
3:20 – 4:40 in Hill-340

or by appointment, in Hill 340

*Home page:*
http://www.math.rutgers.edu/~rwilson

*Telephone contacts:* 445-1317
(Office) and 445-2390 (Math Department)

*In class (midterm)
examinations: *Monday,
October 15; Monday, November 19

*Final examination: *Wednesday
December 19, 12 -3 PM. Location is known a few weeks before the end of term,
and will be announced in class.

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 and some of Chapter 9). Roughly the last third of the course will be devoted to the study of groups (Chapter 7 and some of Chapter 8).

Here it the tentative schedule for the course

September 5 – Lecture on Chapter 1; Workshop #1

September 10 – Lecture on Chapter 2

September 12 – Lecture on Sections 3.1 and 3.2; Workshop #2

September 17 – Lecture on Section 3.3

September 19 – Lecture on Section 6.1; Workshop #3

September 24 – Lecture on Section 6.2

September 26 – Lecture on Section 4.1; Workshop #4

October 1 – Lecture on Sections 4.2, 4.3

October 3 – Lecture on Section 4.4; Workshop #5

October 8 – Lecture on Sections 4.5, 4.6

October 10 – Review; Workshop #6

October 15 – Exam #1

October 17 – Lecture on Sections 5.1, 5.2; Workshop #7

October 22 – Lecture on Section 5.3

October 24 – Lecture on Section 6.3; Workshop #8

October 29 – Lecture on Section 9.1

October 31 – Lecture on Section 9.2; Workshop #9

November 5 - Lecture on Section 9.3

November 7 - lecture on Sections 7.1, 7.2; Workshop #10

November 12 – Lecture on Sections 7.3, 7.4

November 14 – Review; Workshop #11

November 19 – Exam #2

November 21 – NO CLASS (Friday schedule)

November 26 – Lecture on Section 7.5

November 28 – Lecture on Sections 7.6, 7.7; Workshop #12

December 3 – Lecture on Sections 7.8

December 5 – Lecture on Section 7.9, 7.10; #Workshop #13

December 10 – Lecture on Sections 8.1, 8.2, 8.3

December 12 – Review; Workshop #14

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 (2^{1/2} 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. Plagiarism ^{1 :}**

The workshop involves a mixture of two very different kinds of work:

**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.

^{1}*Acknowledgement:* Substantial portions of this page are adapted or copied from
Professors Sims',
Lyon's and Cherlin’s 351 pages
from the period 2001-2006.