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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