Sample Syllabus


Catalog Description:
Topics include basic discrete probability, including urn models and random mappings; a brief introduction to information theory; elements of number theory, including the prime number theorem, the Euler phi function, the Euclidean algorithm, and the Chinese remainder theorem; and elements of abstract algebra and finite fields including basic fundamentals of groups, rings, polynomial rings, vector spaces, and finite fields. Carries credit toward the Applied Mathematics degree only when followed by CS 579. Recommended for high-level undergraduate students.

Textbook(s): None.

Required: Lecture Notes ( 6 modules )

Recommended: None.


Week-By-Week

Week
 Topics Covered
 Reading
 Assignments
1
 Divisibility in Z , Division algorithm , Euclidean algorithm
 Mod I pp1-7
 Exers in mod
2
 Extended Euclidean algorithm , Prime #s and Prime #theorem , Euler phi function
 Mod I pp 8-14
 "               "
3
 Basic abstract algebra , Mod n arithmetic , Ring of integers mod n, Chinese remainder theorem
 Mod II pp1-7
 "               "
4
 Finite groups , Euler's and Fermat's theorems , Square and multiply algorithm
 Mod II pp 8-14   
"               "
5
 Finite cyclic groups , Cyclic mod n units
 Mod III
"               "
6
 Midterm

7
 Introduction to quadratic residues
Mod IV pp1-7  
"               "
8
 Quadratic residues and the Legendre symbol, Gauss Reciprocity theorem for the Legendre symbol
Mod IV pp 8-15
 "               "
9
 Quadratic residues and the Jacobi symbol , Gauss Reciprocity theorem for the  Jacobi symbol
 Mod IV pp 16-28
 "               "
10
 Polynomial rings , Division algorithm for polynomials , Quotient rings and fields
 Mod V
"               "
11
 Finite fields , Characteristic of a finite field , Vector spaces and finite fields
 Mod VI pp 1-6   
"               "
12
 Minimal polynomials , Uniqueness of finite fields
 Mod VI pp 7-12
 "               "
13
 "                                                             "                                                       
"                      "
 "               "
14
 Mobius inversion and the existence of finite fields
 Mod VI pp 13-20
 "               "