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.
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Course Learning Outcomes
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Learning Outcomes |
Program Learning Outcomes |
Teaching Methods |
Assessment Methods |
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1) Knows the basic properties of divisibility, prime numbers and the fundamental theorem of arithmetic. |
2,4 |
1 |
A,B |
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2) Using Euclidean algorithm, computes the greatest common divisior of integers and the least common multiple of integers. |
2,4,7 |
1 |
A,B |
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3) Solves congruence equations including systems of congruence equations by applying Chinese remainder theorem. |
1,2,4,7,9 |
1 |
A,B |
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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 |
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5) Applies Gauss’ quadratic reciprocity law. |
1,2,3,4,7,9 |
1 |
A,B |
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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
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COURSE CONTENT |
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Week |
Topics |
Study Materials |
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1 |
Divisibility, the greatest common divisor and the least common multiple, primes, unique factorization and the fundamental theorem of arithmetic. |
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2 |
Congruences, Fermat’s Little Theorem, Euler’s Formula. |
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3 |
Euler’s Phi Function and the Chinese Remainder Theorem. |
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4 |
Counting Primes. Euler’s Phi Function and Sums of Divisors. |
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5 |
Arithmetical Functions, Mobius inversion formula. |
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6 |
The structure of the unit group of Zn. |
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7 |
Gauss’ Quadratic Reciprocity. |
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8 |
Arithmetic of quadratic number fields |
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9 |
Pell’s equation |
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10 |
Quadratic Gauss sums |
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11 |
Finite fields. |
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12 |
Gauss and Jacobi sums |
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13 |
Equations over finite fields. |
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14 |
The zeta function and Weil conjectures. |
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Recommended Sources
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Textbook |
A Classical Introduction to Modern Number Theory, K. Ireland, M. Rosen, Graduate Texts in Math., Springer-Verlag. |
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Additional Resources |
Material Sharing
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Documents |
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Assignments |
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Exams |
Assessment
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IN-TERM STUDIES |
NUMBER |
PERCENTAGE |
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Mid-terms |
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Quizzes |
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Assignments |
7 |
100 |
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Total |
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100 |
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CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE |
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40 |
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CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE |
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60 |
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Total |
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100 |
Course’s Contribution to Program
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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) |
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x |
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2 |
Acquiring fundamental knowledge on fundamental research fields in mathematics |
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x |
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3 |
Ability form and interpret the relations between research topics in mathematics |
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x |
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4 |
Ability to define, formulate and solve mathematical problems |
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x |
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5 |
Consciousness of professional ethics and responsibilty |
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x |
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6 |
Ability to communicate actively |
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x |
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7 |
Ability of self-development in fields of interest |
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x |
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8 |
Ability to learn, choose and use necessary information technologies |
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x |
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9 |
Lifelong education |
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x |
ECTS
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ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION |
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Activities |
Quantity |
Duration |
Total |
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Course Duration (14x Total course hours) |
14 |
4 |
56 |
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Hours for off-the-classroom study (Pre-study, practice) |
14 |
3 |
42 |
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Mid-terms (Including self study) |
|
|
|
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Quizzes |
|
|
|
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Assignments |
7 |
5 |
35 |
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Final examination (Including self study) |
1 |
14 |
14 |
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Total Work Load |
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148 |
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Total Work Load / 25 (h) |
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5,92 |
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ECTS Credit of the Course |
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6 |