MAT 312/AMS 351
Applied Algebra


Sorin Popescu (office: Math 3-109, tel. 632-8255, e-mail sorin at
Office hours: Tu 11:30am-12:30pm, Th 2:20pm-3:30pm

Graduate Assistants:

Tanvir Prince (office: Math 2-121, email prince at
Office hours: Tu 6pm-8pm MLC, Wed 1pm-2pm Math 2-121

Travis Waddington (office: Math 3-122, e-mail ratatosk at
Office hours: Tu 1:45-3:45pm MLC, Mo 12-2:00pm 3-122


Lecture: TuTh 12:50pm-2:10pm, Library W0512
Recitation 1: Tu 3:50pm-4:45pm, Physics P117
Recitation 2: Mo 11:45am-12:40pm, Physics P112
Midterm Review session: Monday, Feb 21, 5:00-6:30pm in Math P-131
Midterm Review session: Tue, Mar 29, 6:30-8:00pm in Math P-131
Final exam review session: Wed, May 11, 3:00-5:00pm in Math P-131
Final exam: Tuesday, May 17, 11:00-1:30pm in Old Eng 145

Grades are now posted on the Solar system. Have a nice summer!


Either MAT 203 or MAT 205 or AMS 261 (Calculus III), and MAT 211 or AMS 210 (Linear algebra) are prerequisites for this class. In general basic linear algebra exposure is required and assumed, but I will try to keep prerequisites to a minimum.


Numbers, Groups and Codes, J. F. Humphreys, M. Y. Prest, (second edition), Cambridge University Press.
Available from the university bookstore or check prices at AddAll.

The class is an introduction to algebraic structures and applications. The above textbook moves from algebraic properties of integers, through other examples, to the beginnings of group theory. Applications to finite state machines, public key cryptography and to error correcting codes are also emphasised. Attention is also paid to the historical development of the mathematical ideas presented. The text should be easily accessible for both students of mathematics and computer science.

Here are a number of other good undergraduate books that you may perhaps find useful to consult during the semester (all of them available in our library):

Course description:

We will basically cover the following chapters in the textbook but the schedule below may/will be adjusted based on students' preparation and progress.

TopicSections in textbookWeekNotes
Euclidean division algorithm, GCD/LCM, InductionSections 1.1 and 1.21/24-1/30
Prime numbers, Unique factorization, Congruences, Linear CongruencesSections 1.3 and 1.41/31-2/6
Fermat's theorem, Euler's theorem, applicationsSections 1.5 and 1.62/7-2/13
Public key criptography, Cryptographic protocolsSection 1.62/14-2/20
Sets, functionsSections 2.1 and 2.22/21-2/27Midterm 2/22
RelationsSection 2.32/28-3/6
PermutationsSection 4.13/7-3/13
Order and signature of a permutation, transpositions and cyclesSection 4.23/14-3/20
Groups: definition and examplesSection 4.33/28-4/3Midterm 3/31
Algebraic structuresSection 4.44/4-4/10
Order of an element, generators, subgroupsSections 5.1 and 5.24/11-4/17
Lagrange's theorem; finite groups of small orderSections 5.2 and 5.34/18-4/24
Error detecting/correcting codesSection 5.44/25-5/1
Error detecting/correcting codes (continued)Section 5.45/2-5/8
Review sessionWed 05/113:00-5:00pmP-131
Final ExamTu 05/1711:00am-1:30pmOld Eng 145

Note: Although students may take both MAT 312 and MAT 313, there is some nontrivial overlap in the material of these two courses.

Projects, Homework & Grading:

Students are encouraged to do an individual special project or participate in a group special project. These could involve a historical report on material of the course, including perhaps a brief oral presentation or learning some topic in algebra not discussed in the course or writing a computer program for some algorithm. The choice of topic and the exact scope of the special project are to be determined after consultation with the instructor and the final form of a proposal must be presented in writing to the instructor.

Homework is an integral part of the course. Problems will be assigned periodically. You should try to solve them by yourself. You should also discuss them with your fellow students and you may work together on each problem set, but what you hand in must be your own writing and you should be able to answer questions about its content. The solutions of homework problems can be discussed (after the due date) in lectures and/or more appropriately in recitation sections. Some of the homework problems will be graded and solutions will be posted on the web. Problems marked with an asterisk (*) are for extra credit.

Late homeworks will not be accepted.

There will be two midterm examinations (on 2/22 and 3/31) and a final exam (on 05/17). All examinations are inclusive in the sense that they will cover all the material studied up to a specified date. The exact area of coverage of each examination will be posted on the web. No calculators, notes, or books, etc., will be allowed during the midterms or final exam. Exam dates and times are not flexible and there will be NO makeup exams.

Your grade will be based on the weekly homeworks (20%), midterms (25% each), and the final exam (30%). The two lowest homework grades will be dropped before calculating the average. A special project and class participation may also contribute (up to 15%) toward the final grade (either as bonus, or as substitute for some of the homework).


The following is a (growing) list of web sites devoted to topics relevant for our class: A number of interesting local links that you are warmly encouraged to explore:

Math Learning Center

The Math Learning Center (MLC), located in Room S-240A of the Math Tower, is an important resource. It is staffed most days and some evenings by mathematics tutors (professors and advanced students). For more information and a schedule, consult the MLC web site.

Special needs

If you have a physical, psychiatric, medical or learning disability that may impact on your ability to carry out assigned course work, you may contact the Disabled Student Services (DSS) office (Humanities 133, 632-6748/TDD). DSS will review your concerns and determine, with you, what accommodations may be necessary and appropriate. I will take their findings into account in deciding what alterations in course work you require. All information on and documentation of a disability condition should be supplied to me in writing at the earliest possible time AND is strictly confidential. Please act early, since I will not be able to make any retroactive course changes.