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.
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 Phifunction, 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 Z_{n}. 

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., SpringerVerlag. 
Additional Resources 
Material Sharing
Documents 

Assignments 

Exams 
Assessment
INTERM STUDIES 
NUMBER 
PERCENTAGE 
Midterms 

Quizzes 

Assignments 
7 
100 
Total 

100 
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE 
40 

CONTRIBUTION OF INTERM 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 selfdevelopment in fields of interest 
x 

8 
Ability to learn, choose and use necessary information technologies 
x 

9 
Lifelong education 
x 
ECTS
Activities 
Quantity 
Duration 
Total 
Course Duration (14x Total course hours) 
14 
3 
42 
Hours for offtheclassroom study (Prestudy, practice) 
14 
3 
42 
Midterms (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 