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 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 |
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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 |
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13 |
Equations over finite fields. |
|
14 |
The zeta function and Weil conjectures. |
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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 |
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Assignments |
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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 |
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No |
Program Learning Outcomes |
Contribution |
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1 |
2 |
3 |
4 |
5 |
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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 |
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|
x |
|
|
7 |
Ability of self-development in fields of interest |
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|
|
|
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 |
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Activities |
Quantity |
Duration |
Total |
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 |