Uludağ Üniversitesi, Fen-Edebiyat Fakültesi Matematik Bölümü, Görükle Bursa-TÜRKİYE 0 224 294 17 70 / betulgezer@uludag.edu.tr
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Website:
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Objective of the Course:
The algebraic number theory brings two important areas of mathematics such as algebra and numbery theory. Our first aim is to introduce fundamental ideas of algebraic numbers and the second is to illustrate how basic notions from the theory of algebraic numbers may be used to solve problems in number theory. The main focus is to extend properties of the integer numbers to more general number structures: algebraic number fields and their rings of algebraic integers. Then give an introduction to Fermat’s last theorem. So students can see how basic ideas are used to solve problems in number theory.
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Contribution of the Course to Professional Development
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Learning Outcomes:
1
Learns basic concepts on symetric polynomials, modules, free abelian groups.;
2
Learns algebraic numbers, algebraic integers, integral bases, norms and traces. ;
3
Learns factorization into irreducibles, trivial factorizations and Euclidean domains.;
4
Learns ideals, the decomposition of ideals, the norm and classes of ideals, factorization in cyclotomic fields and lattices.;
5
Learns Minkowski theorem, two and four square theorem.;
6
Learns class groups, finiteness of the class groups and number-theoric applications and some class number calculations.;
7
Learns elliptic curves and the group structure on elliptic curves, Fermat’s last theorem.;
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Course Content:
Week
Theoretical
Practical
1
Basic concepts on groups, ring and fields and some elementary theorems.
Algebraic numbers, algebraic integers, integral bases, norms and traces.
4
Rings of integers, quadratic and cyclotomic fields.
5
Factorization into irreducibles, trivial factorizations and Euclidean domains.
6
Ideals, the decomposition of ideals.
7
The norm and classes of ideals.
8
Factorization in cyclotomic fields and lattices.
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Minkowski theorem, two and four square theorem.
10
Class groups, finiteness of the class group.
11
Factorization of elements in an extension ring.
12
Number-theoric applications and some class number calculations.
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Elliptic curves and the group structure on elliptic curves.
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Overview on Fermat’s last theorem.
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Textbooks, References and/or Other Materials:
[1]Algebraic Number Theory and Fermat’s Last Theorem, Ian Stewart, David Tall. [2]Algebraic Numbers, Paulo Ribenboim. [3]Introductory Algebraic Number Theory, Ş. Alaca, K.S. Williams.
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Assesment
TERM LEARNING ACTIVITIES
NUMBER
PERCENT
Midterm Exam
1
40
Quiz
0
0
Homeworks, Performances
0
0
Final Exam
1
60
Total
2
100
Contribution of Term (Year) Learning Activities to Success Grade
40
Contribution of Final Exam to Success Grade
60
Total
100
Measurement and Evaluation Techniques Used in the Course
Information
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ECTS / WORK LOAD TABLE
Activites
NUMBER
TIME [Hour]
Total WorkLoad [Hour]
Theoretical
14
3
42
Practicals/Labs
0
0
0
Self Study and Preparation
14
5
70
Homeworks, Performances
0
0
0
Projects
0
0
0
Field Studies
0
0
0
Midtermexams
1
15
15
Others
14
1
14
Final Exams
1
9
9
Total WorkLoad
150
Total workload/ 30 hr
5
ECTS Credit of the Course
5
26
CONTRIBUTION OF LEARNING OUTCOMES TO PROGRAMME QUALIFICATIONS