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
1)	Be able to organize events, for the development of critical and creative thinking and problem solving skills, by using appropriate methods and techniques.
2)	Ability to make individual and team work on issues related to working and social life.
3)	Ability to transfer ideas and suggestions, related to topics about his/her field of interest, written and verball.
4)	Ability to use mathematical knowledge in technology.
5)	To apply mathematical principles to real world problems.
6)	Ability to use the approaches and knowledge of other disciplines in Mathematics.
7)	Be able to set up and develope a solution method for a problem in mathematics independently, be able to solve and evaluate the results and to apply them if necessary.
8)	To be able to link abstract thought that one has to concrete events and to transfer the solutions and examine and interpret the results scientifically by forming experiments and collecting data.