Contents |
Course Code
IAM 519 (9700519)
Credit
(4-0) 4
Prerequisites
Consent of the instructor
Content/Aims
The main objective of this course is to prepare students for later studies in Cryptography Graduate Program of IAM, and also to explain some problems that are easy to ask but still unsolved and to give some ideas about why abstractions are to be made, by giving fundamental properties of integers and some algebraic preliminaries.
The content of the course is as follows: Basic properties of Integers, Divisibility, Primes, The fundamental theorem of arithmetic, Fermat numbers, Factorization methods, Diophantine equations, Congruences, Theorems of Fermat, Euler and Wilson, Arithmetical functions, Primitive roots, Quadratic congruences, Group, Field, Field extensions, Finite fields, Factorization of polynomials, Splitting field.
Learning Outcomes
Suggested Textbooks/Resources
- K. H. Rosen, Elementary Number Theory and its Applications, Addison –Wesley, 1992
- David M. Burton, Elementary Number Theory, The McGraw-Hill, 1998
- R. Lidl and H. Niederreiter, Finite Fields, Cambridge Univ. Pres, 1986
- A. Adler and J. E. Coury, The Theory of Numbers, Jones and Bartlett Publishers, 1995
Outline
- The Integers: Basic Properties, Mathematical induction, Binomial coefficients, Divisibility, Representataions of integers, Prime numbers, Greatest common divisor, Euclidean algorithm, The fundamental theorem of arithmetic, Fermat numbers and factorization methods, Linear diophantine equations, (5 weeks)
- Congruences : linear congruences, The Chinese remainder theorem, Systems of linear congruences, Applications of congruences. Theorems of Fermat, Euler and Wilson. Arithmetical functions, Primitive roots, Quadratic residues, Quadratic reciprocity, (5 weeks)
- Algebraic Preliminaries : Groups, Fields, Polynomials,Field extensions, Finite fields, Factorization of polynomials, Factorization of polynomials, Splitting field of a polynomial, multiple roots, (5 weeks)