Math 401-402 (2) Modern Algebra SYLLABUS

Teacher: Visiting Professor Robert Bieri

e-mail: 1) , 2)

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 discuss questions with friends, classmates and teachers. Communication is indeed a most important ingredient of this. Thus students are strongly encouraged to ask questions and trigger discussions in or after class – but be warned: Learning mathematics is a slow procedure; it would be a fatal mistake to think one can squeeze it into concentrated learning periods in the days before an exam.

    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.

    Textbook and Course Contents

    ``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 Schedule

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

The problems of Test 1

    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

The problems of Test 2

    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
polynomial rings)

The problems of Test 3

     Final Exam:  December 19th 08:30 - 10:30   Room  SL 210
    (Special session for  Emiko Lee Okamoto and Inesa Anastas Ziu:
December 20th 09:40 - 11:40   Room  LN 2201).
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.

    For more information see Prof. Feingold's home page.


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
Typical Test Type Questions II
Typical Test Type Questions III
Summary of important definitions on groups

Timetable for the Tutorials

IMPORTANT: NO LECTURES AND TUTORIALS on December 4th, 5th and 6th
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