To the page of Algebra I MAT534
MAT 312 - AMS 351: Applied Algebra

Stony Brook, Fall 2005
Last update: December 17, 2005

This page will be updated at least once a week. Please check it regularly for announcements.

Syllabus Course description Last homework Week by week details Grading Projects

Important announcement: The classroom for the final exam has been CHANGED. The final exam will be held in Engineering 143 on December 20, Tuesday 11am-1:30pm. Practice problems for the exam and selected solutions are posted.
About optional projects: The deadline for the projects has been extended to December 22. You can e-mail your projects or bring them to my office by noon December 22. This is an absolute deadline. Late projects will not be accepted after this deadline under any circumstances.

Classes: TuTh 11:20am-12:40pm, Hvy Engr Lab 201
Instructor: Valentina Kirichenko (office Math 3-102, phone 632-8884)
Webpage: http://www.math.sunysb.edu/~vkiritch/
Office Hours: Tu 3-5, Th 2-3 (Math 3-102), Th 3-4 (MLC) and by appointment
Recitation 1: Tu 12:50pm- 1:45pm Physics P117
Recitation 2: W 11:45am-12:40pm Physics P112
Teaching assistant: Luis Lopez
Webpage: http://www.math.sunysb.edu/~llopez
Office Hours: M 5-7 (Math-2-122), M 7-9 (Math-S240A)

About this Course: This course is an introduction to algebra with emphasis on concrete examples. We start with elementary number theory and its application to public key codes. Then we study elementary set theory. We continue with foundations of logic, some proof strategies and logical calculus. The rest of the course will be devoted to group theory and error-correcting codes. If time permits we will also discuss polynomials, finite fields and cyclic codes. Number theory

Number theory Prerequisites: MAT 203 or MAT 205 or AMS 261; MAT 211 or AMS 210
Textbook: Numbers, Groups and Codes, J. F. Humphreys, M. Y. Prest, (second edition), Cambridge University Press
To buy this book check AddAll.		 Number theory
Number theory Additional reading: All books listed below are available in Math/Physics library

A Friendly Introduction to Number Theory, J.H. Silverman,

What is Mathematics? An elementary approach to ideas and methods, Richard Courant and Herbert Robbins,

Number theory. An approach through history, A. Weil.

Disquisitiones arithmeticae, K.F. Gauss

Grading:   There will be one midterm test, 80 minutes long, given in class - no makeups. If it is missed because of a serious (documented) illness or emergency, the semester grade will be determined based on the balance of the work in the course. A final examination will be held on December 20, Tuesday 11am-1:30pm. Students are expected to ensure when they register for this course that they will be available for the final examination, and that they do not have too many final exams on that date. The final course grades will be determined as follows:

Homework/Class Participation 30%, Midterm Test 30%, Final Exam 40%

 Number theory
Homework/Class Participation: You can not learn mathematics without doing mathematics. It is essential to be an active participant in class and to solve problems.
Some statements used in class will be left as exercises. These exercises will be posted further down on this page and will be discussed in the next class. You can volunteer to explain (on the blackboard) your solution of such an exercise to the others in class for extra credit.
Each week a homework assignment will be posted further down on this page, normally on Wednesday. It is due the following week by Thursday noon in class or directly with our teaching assistant. While you may work together with others in the class (which can be a rewarding experience), write up your own solutions in your own words. Since homework earns credit, it is assumed that everyone submitting particular problems has solved them individually. The goal of the homework is to understand the material, not to merely hand in some paper. This is a more advanced math course where coherent arguments and rigorous proofs are often required. Late homework will not be accepted.
Syllabus:
Weeks Sections
August 30, September 1 1.1
September 6, 8 1.2, 1.3
September 13, 15 1.4
September 20, 22 1.5, 1.6
September 27, 29 1.6
October 6 2.1, 2.2
October 11 2.2
October 18, 20 2.3, overview of 3.1-3.3
October 25,27 4.1
November, 1 Review session for the midterm
November, 3 Midterm test
November 8, 10 4.2
November 15, 17 4.3, overview of 4.4, 5.1, beginning of 5.3
November 22 5.2
November 29, December 1 5.4
December 6, 8 5.4
December 13 Review session
Special Needs: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room 128, (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.
Students requiring emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information, go to the following web site.
http://www.ehs.stonybrook.edu/fire/disabilities.asp
Week by Week Details, Exercises and Homework Assignments:

August 30, September 1
Division with residue, notions of divisor and common divisor, Euclidean algorithm for the greatest common divisor of two numbers.
Exercise for class participation: Prove that any common divisor of two integers divides their greatest common divisor.
Homework 1 (due September 8)

September 6,8
Divisibility criteria by 3, 9 and 11. Mathematical induction. Unique factorization theorem.
Exercise for class participation: Prove that for any integers m, d and k, the integers m and m+kd have the same residue under division by d.
Homework 2 (due September 15)

September 13,15
Congruence classes modulo n. Inverses for congruent classes.
Exercise for class participation: Write addition and multiplication tables for the congruence classes modulo 3.
Homework 3 (due September 22)

September 20,22
Chinese Remainder Theorem, Fermat's Little Theorem
Exercise for class participation: Let m and n be relatively prime integers. Prove, that if x and y are congruent both modulo m and modulo n, then x and y are congruent modulo mn.
Homework 4 (due September 29)

September 27,29
Euler's Theorem. Public key codes.
No exercise for class participation this week:
Homework 5 (due October 6)

October 6
Sets, functions.
No exercise for class participation this week:
Homework 6 (due October 18 (Tuesday))

October 11
Functions, cardinality of sets.
Exercise for class participation this week: Show that there exists a surjection from the set of n elements to the set of m elements if and only if n is greater than or equal to m.
Homework 7 (due October 20)
Solutions

October 18,20
Relations. Logic. Truth tables. Calculus of propositions.
No homework. Prepare for the midterm.

October 25,27
Groups of permutations: product, inverses, cycle decomposition and order of permutations.
Homework 8 (due November 10)
Midterm program.
The midterm will cover Sections 1.1-1.6. Calculators, books and notes are not allowed. Topics include Euclidean algorithm for the greatest common divisor, prime decomposition, inverses to congruence classes, Chinese remainder theorem, Euler function, Fermat's and Euler's theorems. The midterm will not include public key codes. For the midterm, you will also need to be familiar with the notions introduced in Sections 2.1-2.2 and 3.1-3.3. Sets, intersection and union of sets, functions, conjunction, disjunction and negation of propositions, implication may be used in formulation of some midterm problems.
  • Example 1: Let A be the set of all integers a such that a is congruent to 1 modulo 5, and let B be the set of all integers a such that a is congruent to 4 modulo 7. Find the intersection of A and B.
  • Example 2: Is the following statement true or false?
    If n is a multiple of 7 and n is congruent to 2 modulo 5, then n is congruent to 7 modulo 35.
Relations will not appear in the test. Problems of the midterm will be, in essence but not in form, similar to the first five problems of Homework 7. For more practice solve Problems 1.1.2, 1.2.3, 1.3.9, 1.4.3 (iv,v), 1.5.2, 1.6.6.
November 8, 10
Sign of a permutation. Permutations in solving algebraic equations.
Homework 9 (due November 17)
This is not a part of the homework, but just a game: Fifteen puzzle

November 15, 17
Definition and examples of groups. Algebraic structures. Order of an element. Isomophisms.
Homework 10 (due November 29 (Tuesday))

November 22
Lagrange theorem. Cyclic groups.
Homework 11 (due December 6)

November 29, December 1, 6, 8
Error-correcting codes.
Homework 12 (due December 13)

Projects. There are optional projects for this course. You can submit your work on no more than one of such projects for extra credit. Each project gives up to 10% of the course grade. Here is a list of possible topics for the projects. Other projects include independent study of topics (such as set theory) that were not fully covered in the course. The choice of a topic and the exact scope of the special project are to be determined after consultation with the instructor. The deadline to select a project is November 3 (you should inform me about your choice via e-mail). The deadline for submitting projects is December 20.