Math 75
Applied topics in number theory and algebra: finite fields and coding theory
Instructor: Carl Pomerance (carl.pomerance at dartmouth.edu)
Abstract  Classes  Tutorials  Staff  Textbook  Grading  News and current assignment  Past assignments  Exams  Honor Code
News and current assignment 
If you'd like to drop by my office early this week I can return your recent assignments. There will be no final examination. Here is an article of mine on primality testing written a few years ago. Here are some notes I prepared on finite fields. I hope they're useful.And here is a continuation of the notes. Our text is "Errorcorrecting codes and finite fields" by Oliver Pretzel, published by Oxford. This book should be at Wheelock Books, and it is available on amazon and other sites. I taught this class (jointly with Paul Pollack) in Spring term 2008, covering the same topics. You might find the class notes from that term helpful; they're here.  
Abstract 
Finite fields and coding theory The principal mathematical tool behind errorcorrecting codes is the theory of finite fields. We have all run into fields, even if we haven't taken a course in algebra. For example, the field of rational numbers, the field of real numbers, and the field of complex numbers: these should all be familiar. But they are all infinite fields. A finite field enjoys the same axioms of addition and multiplication as these infinite fields, but it has only finitely many elements. A "prime" example: the integers mod p, where p is a prime number. There are also somewhat more complicated examples that are built off of these "prime" cases. We shall see in this course how to exploit the algebra of finite fields to consruct efficient errorcorrecting codes. Prerequisites: An undergrad number theory course and/or an undergrad course in abstract algebra. A lot of what we'll be doing involves some linear algebra. I'll be happy to try and fill in gaps for motivated students. Grading: There will be weekly written assignments, a midterm, and a final exam. Enrolled graduate students who have been admitted to candidacy: please see me about requirements.  
Classes 
Room: 004 Kemeny We may meet several of the xhours, but this will always be announced in advance.  
Staff 


Textbook 
See above.  
Homework 
Homework is due at the start
of the class period on the due date.


Past assignments 
Homework assignment 1 is here. The attached assignment is due on Wednesday, April 9. The attached assignment is due on Monday, April 14. Here is the
takehome midterm due at the start of class
on Monday, April 21. Assignment due Monday, April 28: Chapter 3 in the text, numbers
7, 8, 10, 16, 17, 19. (Note that "error pattern" is defined in the
next chapter.) Homework assignment due Monday, May 5, 2014: Homeword due Wednesday, May 14: Ch.14, #2, #4, #5#10. Homework due Wednesday, May 21.
This assignment is due Wednesday, May 28. 

Exams 
There will be a midterm and a final exam.  
Grading 
Grades will be based on homework and exams. 

Honor Code 
Collaboration on homework is definitely allowed and even encouraged. However, it is tempting to think that you understand something that was figured out by your friend. When you hand in a solution, you should know it well enough that you could explain it to others. Please name others you worked with when handing in homework papers. Merely copying (electronic or otherwise) of another person's solutions, in whole or in part, is a violation of the Honor Code, even if attribution is made. You should understand what you turn in. 

Disabilities 
I encourage any students with disabilities, including "invisible" disabilities such as chronic diseases and learning disabilities, to discuss appropriate accommodations with me, which might help you with this class, either after class or during office hours. Dartmouth College has an active program to help students with disabilities, and I am happy to do whatever I can to help out, as appropriate. The Student Disabilities Coordinator, Nancy Pompian, can be reached at 62014 if you have any questions. Any student with a documented disability requiring academic adjustments or accommodations is requested to speak with me by the end of the second week of the term. All discussions will remain confidential, although the Academic Skills Center may be consulted to verify the documentation of the disability and advise on an appropriate response to the need. It is important, however, that you talk to me soon, so that I can make whatever arrangements might be needed in a timely fashion. 