MAT6002 Advanced Algebra IIBahçeşehir UniversityDegree Programs MATHEMATICS (TURKISH, PHD)General Information For StudentsDiploma SupplementErasmus Policy StatementNational Qualifications
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 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