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
| COURSE CONTENT | ||
| 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’s Contribution to Program
| 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 fundamental 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
| ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION | |||
| Activities | Quantity |
Duration (Hour) |
Total Workload (Hour) |
| Course Duration (14x Total course hours) | 14 | 4 | 56 |
| 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 | 14 | 14 |
| Total Work Load | 148 | ||
| Total Work Load / 25 (h) | 5,92 | ||
| ECTS Credit of the Course | 6 | ||