Teacher: Visiting Professor Robert Bieri

e-mail: 1) bieri@math.uni-frankfurt.de , 2) rbieri@math.binghamton.edu

Office: LN-2247, Phone: 777-4263, Office Hours: MWF 11:00 - 1:00 and by appointment.

Class meets: MWF 9:40 - 10:40 in SW 324, Thursday in 11:40 - 1:05
in FA 244.

** How to Learn Mathematics**

Experience shows that one cannot learn
mathematics by only reading a book or listening to the lectures.
Reading and listening has to be combined with more active periods
of **doing exercises**, **solving problems**, **writing
down a proofs**, **drawing schematic pictures** (as a
visual bridge to understanding examples and abstract concepts) and
**looking for counter examples** (in order to understand the
borderlines of a core idea). Class attendance is absolutely
essential as the first introduction to the theoretical material,
and it is necessary to understand the theory in order to do
sensible calculations and interpret them correctly. But class
attention is only the first step – the best (and perhaps the only)
way to learn and internalize the content of a lecture is to **take
notes****,** * compare them with the literature*
and

Learning mathematics requires intense concentration and considerable effort, and only few are able to it without the internal motivation of a genuine interest in its contents: in the beauty of the arguments, in the power of the applications, in the security of unquestionable truth, or something along these lines.

``Modern Algebra, Notes for Math 401-402 at
Binghamton University'',1st Edition by Matthew G. Brin, which is
available here.

In the first semester we will cover
introductory material on various algebraic structures (groups,
rings, fields, pages 91-114), and then material on group theory
(Chapter II, pages 117-198). In the second semester, Professor
Alex Feingold will continue in this book with ring theory, field
theory and Galois theory (Chapter III, pages 199-290). Additional
topics may be covered if there is time and interest.

There will be 3 ``hourly'' exams (actually 1 hour and 25 minutes) administered during the Thursday meeting, and 1 Final Exam during the scheduled Finals period. The hourlies will be worth 100 points each, and the (2-hour) Final Exam will be worth 150 points. The contents of each exam will be determined one week before the exam. The Final Exam will be comprehensive, covering the whole course. ANYONE UNABLE TO TAKE AN EXAM SHOULD CONTACT THE PROFESSOR AHEAD OF TIME TO EXPLAIN THE REASON. A MESSAGE CAN BE LEFT AT THE MATH DEPT OFFICE. NO ONE SHOULD MISS THE FINAL!

** Exam 1: September 27th. Contents:
Sets and maps, Permutations, Groups: Definition, elementary
consequences and Examples (the symmetric permutation groups,
dihedral group, Integers mod m)**

Exam 2: October 25th. Contents are the
Sections

II. 3.__ Integers mod m__ (*additive* and *multiplicative*
group, the field Z/pZ ). Text 64 - 74.

II. 4. __Subgroups __ (with examples from*
permutation groups*, *dihedral groups*, and and *cyclic
groups*). Text 95-97, 161- 165

II. 5. __Cosets and Factor Groups__ (including*
conjugation* -- in general and in the symmetric permutation
group) Text 167 - 174

Partly II. 6. __Homomorphisms__ (The definition with kernel,
image, isomorphisms and the Isomorphism Theorem) Text 97-100,
174 -175.

See also the key words and references in Summary
of important definitions on groups

**Exam 3:** **November 29th. Contents
are the Sections
II.6. Group Homomorphisms (The isomorphism
theorem)
III.1. Rings: Definition and Examples (The Polynomial
ring)
III.2. Ring-homomorphisms Ideals and the quotient-ring
III.3. Special ideals (Ideals generated by elements,
principal ideals, prime ideals, maximal ideals)
III.4. Divisibility in integral domains (units,
irreducible elements, prime elements.
III.Principal ideal domains (gcd, prime elements and ideals, and
field quotients of**

**Final Exam: December 19th
08:30 - 10:30 Room SL 210
(Special session for Emiko Lee
Okamoto and Inesa Anastas Ziu: **

Contents: The topics of Tests 1 - 3

Exams will be a combination of theory questions (proofs) and calculations appropriate for a course of this level. For each exam the numerical score will be given a letter grade interpretation, giving each student a letter grade as well as a number grade, and the Total of all points earned will also be interpreted. The letter grades on the exams indicate how a student is doing, and will be taken into consideration when interpreting the Totals. The course grade will be determined by the Total points earned. Only borderline cases will be subject to further adjustment based on Homework. Any cases of cheating will be subject to investigation by the Academic Honesty Committee of Harpur College.

For each section of material covered there will be an assignment of problems from the textbook. They will be due one week from the day they are assigned (or the next scheduled class meeting after that if there is a holiday). Late assignments will be accepted at the discretion of the Professor. Assignments will be examined by a grader, who will record the fact that an assignment was attempted, and give some feedback on how selected problems were done. Lectures can be interrupted at any time for questions. At the start of each class be ready to ask (or answer!) questions about the previous lecture and on homework problems. Although homework will not be precisely graded, the number of attempted homework problems and the quality of the attempts will be considered as a factor in determining your course grade if you are a borderline case for the interpretation of the Total score.

LINK TO THE ASSIGNMENTS

Assignment for the week Sept. 5th -- 12th

Assignment for the week Sept. 12th-- 19th

Assignments for the week Oct. 3rd -- 10th

Assignments for the week Oct. 10th -- 17th

Assignments for the Thanksgiving week

Instructive exercises on sets and functions

Typical Test Type Questions II

Typical Test Type Questions III

Summary of important definitions on groups

Timetable for the Tutorials

The next lecture is on Friday, December 7th, and the assignment for this day is: A neat and complete written solution of all problems of Test III