BAHÇEŞEHİR UNIVERSITY BİLGİ PAKETİ / DERS KATALOĞU
| MATHEMATICS (TURKISH, PHD) | |||||
| PhD | TR-NQF-HE: Level 8 | QF-EHEA: Third Cycle | EQF-LLL: Level 8 | ||
Course Introduction and Application Information
| Course Code | Course Name | Semester | Theoretical | Practical | Credit | ECTS |
| MAT6002 | Advanced Algebra II | Fall Spring |
3 | 0 | 3 | 8 |
| This catalog is for information purposes. Course status is determined by the relevant department at the beginning of semester. |
Basic information
| Language of instruction: | Turkish |
| Type of course: | Departmental Elective |
| Course Level: | |
| Mode of Delivery: | Face to face |
| Course Coordinator : | Assoc. Prof. ATABEY KAYGUN |
| Recommended Optional Program Components: | None |
| Course Objectives: | To provide the necessary algebraic tools and techniques to a graduate student who would go into a doctoral program in mathematics. |
Learning Outcomes
|
The students who have succeeded in this course; A student who finished this course successfully will have acquired the necessary basic knowledge to be able to follow a course in algebra in a doctoral program in mathematics. |
Course Content
| Polynomial algebras over fields, their ideals and quaotients. Euclidean division algorithm. Free monoids. Lexicographical ordering. Other monomial orderings. Buchberger algorithm and Groebner bases. Symmetry groups of field extensions and Galois extensions. The Fundamental Theorem of Algebra. Algebraic closure. Seperable closure. Transcendental extensions and transcendence degree. Krull dimension of an algebra. Noetherian algebras and finite generation. Nilpotent elements, nilpotent and nil ideals. Radicals. Nullstellensatz. Affine varieties. Zariski topolojisi. İndirgenemeyen alt uzaylar. |
Weekly Detailed Course Contents
| Week | Subject | Related Preparation |
| 1) | Polynomial algebras over fields, their ideals and quaotients. Euclidean division algorithm. | |
| 2) | Free monoids. Lexicographical ordering. Other monomial orderings. | |
| 3) | Buchberger algorithm and Groebner bases. | |
| 4) | Buchberger algorithm and Groebner bases. | |
| 5) | Irreducible polynomials and field extensions. | |
| 6) | Symmetry groups of field extensions and Galois extensions. | |
| 7) | Examples from Galois extensions and calculations. | |
| 8) | The Fundamental Theorem of Algebra. Algebraic closure. Seperable closure. | |
| 9) | Transcendental extensions and transcendence degree. Krull dimension of an algebra. | |
| 10) | Noetherian algebras and finite generation. Nilpotent elements, nilpotent and nil ideals. Radicals. | |
| 11) | Nullstellensatz. | |
| 12) | Affine varieties. Examples. | |
| 13) | Zariski topology. Irreducible subvarieties. | |
| 14) | Selected tpoics from affine algebraic geometry. |
Sources
| Course Notes / Textbooks: | Instructor's own lecture notes. S. Lang, "Algebra" |
| References: |
Evaluation System
| Semester Requirements | Number of Activities | Level of Contribution |
| Project | 1 | % 20 |
| Midterms | 1 | % 30 |
| Final | 1 | % 50 |
| Total | % 100 | |
| PERCENTAGE OF SEMESTER WORK | % 30 | |
| PERCENTAGE OF FINAL WORK | % 70 | |
| Total | % 100 | |
ECTS / Workload Table
| Activities | Number of Activities | Duration (Hours) | Workload |
| Course Hours | 14 | 3 | 42 |
| Study Hours Out of Class | 14 | 6 | 84 |
| Project | 1 | 51 | 51 |
| Midterms | 1 | 2 | 2 |
| Final | 1 | 21 | 21 |
| Total Workload | 200 | ||
Contribution of Learning Outcomes to Programme Outcomes
| No Effect | 1 Lowest | 2 Low | 3 Average | 4 High | 5 Highest |
| Program Outcomes | Level of Contribution |