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. ALİ ULAŞ ÖZGÜR KİŞİSEL |
| 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