MATH523 ALGEBRAIC NUMBER THEORY
| Course Code: | 2360523 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Prof.Dr. ÖMER KÜÇÜKSAKALLI |
| Offered Semester: | Fall Semesters. |
Course Objectives
Algebraic number theory originated in attempts to solve Diophantine equations, the most notable of which is Fermat's last theorem, a conjecture that remained open for more than 300 years. The theory is now mature. Through factorizing ideals into products of prime ideals in number fields, one obtains numerous applications to problems in number theory and other fields of mathematics and science. The aim of this course is to introduce the student to this theory and endow him/her with the techniques necessary to perform computations and carry out proofs in this area.
Course Content
Ring of integers of an algebraic number field. Integral bases. Norms and traces. The discriminant. Factorization into irreducibles. Euclidean domains. Dedekind domains. Prime factorization of ideals. Minkowskis theorem. Class-group and class number.
Course Learning Outcomes
By the end of this course, a student will:
- compute rings of integers in various number fields
- carry out proofs about unique factorization into prime ideals
- explicitly find factorizations in low degree examples
- compute some ideal class groups
- compute groups of units using Dirichlet’s theorem
- use Minkowski’s theorem to bound the class number
- solve certain Diophantine equations using properties of the relevant number field
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution