Course Code:
MATH 421
Semester:
Fall
Course Type:
Core
P:
3
Lab:
0
Laboratuvar Saati:
0
Credits:
3
ECTS:
5
Course Language:
English
Course Objectives:
To introduce the fundamental topics in elementary number theory.
Course Content:

Integers, divisibility, prime numbers, congruences,Chinese remainder theorem, arithmetic functions, quadratic reciprocity law, quadratic fields, Pell’s equation, further topics including equations over finite fields, zeta functions and Weil conjectures.

Course Methodology:
1: Lecture, 2: Problem Solving
Course Evaluation Methods:
A: Written examination, B: Homework

## Vertical Tabs

### Course Learning Outcomes

 Learning Outcomes Program Learning Outcomes Teaching Methods Assessment Methods 1) Knows the basic properties of divisibility, prime numbers and the fundamental theorem of arithmetic. 2,4 1 A,B 2) Using Euclidean algorithm, computes the greatest common divisior of integers and the least common multiple of integers. 2,4,7 1 A,B 3) Solves congruence equations including systems of congruence equations by applying Chinese remainder theorem. 1,2,4,7,9 1 A,B 4) Knows the basic properties of Euler’s Phi-function, and arithmetic functions, applies Mobius inversion formula. 1,2,3,4,7,9 1 A,B 5) Applies Gauss’ quadratic reciprocity law. 1,2,3,4,7,9 1 A,B 6) Knows the elementary theory of equations over finite fields and the statements of Weil conjectures. 1,2,3,4,7,9 1 A,B

### Course Flow

 Week Topics Study Materials 1 Divisibility, the greatest common divisor and the least common multiple, primes, unique factorization and the fundamental theorem of arithmetic. 2 Congruences, Fermat’s Little Theorem, Euler’s  Formula. 3 Euler’s Phi Function and the Chinese Remainder Theorem. 4 Counting Primes. Euler’s Phi Function and Sums of Divisors. 5 Arithmetical Functions, Mobius inversion formula. 6 The structure of the unit group of Zn. 7 Gauss’ Quadratic Reciprocity. 8 Arithmetic of quadratic number fields 9 Pell’s equation 10 Quadratic Gauss sums 11 Finite fields. 12 Gauss and Jacobi sums 13 Equations over finite fields. 14 The zeta function and Weil conjectures.

### Recommended Sources

 Textbook A Classical Introduction to Modern Number Theory,  K. Ireland, M. Rosen, Graduate Texts in Math., Springer-Verlag. Additional Resources

### Material Sharing

 Documents Assignments Exams

### Assessment

 IN-TERM STUDIES NUMBER PERCENTAGE Mid-terms Quizzes Assignments 7 100 Total 100 CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE 40 CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE 60 Total 100

 COURSE CATEGORY Expertise/ Field Courses

### Course’s Contribution to Program

 No Program Learning Outcomes Contribution 1 2 3 4 5 1 The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry) x 2 Acquiring fundamental knowledge on fundamental research fields in mathematics x 3 Ability form and interpret the relations between research topics in mathematics x 4 Ability to define, formulate and solve mathematical problems x 5 Consciousness of professional ethics and responsibilty x 6 Ability to communicate actively x 7 Ability of self-development in fields of interest x 8 Ability to learn, choose and use necessary information technologies x 9 Lifelong education x

### ECTS

 Activities Quantity Duration (Hour) Total Workload (Hour) Course Duration (14x Total course hours) 14 3 42 Hours for off-the-classroom study (Pre-study, practice) 14 3 42 Mid-terms (Including self study) Quizzes Assignments 7 5 35 Final examination (Including self study) 1 10 10 Total Work Load 129 Total Work Load / 25 (h) 5.16 ECTS Credit of the Course 5