MATH 431 | Course Introduction and Application Information - Department of Mathematics

Course Name

Field theory

Code	Semester	Theory (hour/week)	Application/Lab (hour/week)	Local Credits	ECTS
MATH 431	Fall/Spring	3	0	3	6

Prerequisites

	MATH 225 To attend the classes (To enrol for the course and get a grade other than NA or W)
and	MATH 226 To attend the classes (To enrol for the course and get a grade other than NA or W)

Course Language

English

Course Type

Elective

Course Level

First Cycle

Mode of Delivery

-

Teaching Methods and Techniques of the Course

Lecture / Presentation

National Occupation Classification

-

Course Coordinator

Dr.Öğretim Üyesi NECLA KOÇHAN

Course Lecturer(s)

Dr.Öğretim Üyesi NECLA KOÇHAN

Assistant(s)

Course Objectives	To enable students to grasp the fundamental properties of Fields and Galois theory.
Learning Outcomes	The students who succeeded in this course; understand the meaning of solving a polynomial equation using radicals. understand field extensions. use simple and algebraic extensions. understand separable extensions. introduce the automorphisms of fields. understand normal extensions. introduce the fundamentals of Galois theory. apply Galois theory introduce primitive element theorem and apply it.
Course Description	The focus of the course will be the fields extensions, separable extensions, field automorphisms, the fundamental properties and applications of Galois theory, and Lagrange's theorem. This course is a complement of abstract algebra and enables students to understand abstract notions as solid structures.
Related Sustainable Development Goals

Course Category	Core Courses	X
	Major Area Courses
	Supportive Courses
	Media and Management Skills Courses
	Transferable Skill Courses

Week	Subjects	Related Preparation
1	Recollections: Definitions of group, ring and field	“Galois Theory“ by David A.Cox, Wiley, 2nd Ed., 2004. ISBN-9781118072059 5. Appendix A.
2	Cubic and quartic equations, Cardan’s formulas.	“Galois Theory“ by David A.Cox, Wiley, 2nd Ed., 2004. ISBN-9781118072059 5. Chapter 1.
3	Symmetric polynomials. Discriminant	“Galois Theory“ by David A.Cox, Wiley, 2nd Ed., 2004. ISBN-9781118072059 5. Chapter 2.
4	Roots of polynomials and the fundamental theorem of algebra	“Galois Theory“ by David A.Cox, Wiley, 2nd Ed., 2004. ISBN-9781118072059 5. Chapter 3.
5	Extension fields: The Degree of an Extension, the Tower theorem	Eugene Don, “Schaum's Outline of Mathematica”, 3rd edn (McGraw-Hill, 2018), chapter 2, Chapter 4.
6	Basic and normal extensions	Eugene Don, “Schaum's Outline of Mathematica”, 3rd edn (McGraw-Hill, 2018), Chapter 5.
7	Separable extensions, Theorem of the Primitive Element	Eugene Don, “Schaum's Outline of Mathematica”, 3rd edn (McGraw-Hill, 2018), chapter 5.
8	Midterm
9	Fundamentals of Galois theory	Eugene Don, “Schaum's Outline of Mathematica”, 3rd edn (McGraw-Hill, 2018), Chapter 6.
10	Abelian equations and applications of Galois theory	Eugene Don, “Schaum's Outline of Mathematica”, 3rd edn (McGraw-Hill, 2018), Chapter 6, Chapter 7.
11	Galois extensions	Eugene Don, “Schaum's Outline of Mathematica”, 3rd edn (McGraw-Hill, 2018), Chapter 7.
12	Solvable groups	Dirk P. Kroese, Thomas Taimre, Zdravko I. Botev, “Handbook of Monte Carlo Methods”, (Wiley, 2011), Chapter 8.
13	Cyclotomic extensions	Dirk P. Kroese, Thomas Taimre, Zdravko I. Botev, “Handbook of Monte Carlo Methods”, (Wiley, 2011), Chapter 9.
14	Finite fields	Dirk P. Kroese, Thomas Taimre, Zdravko I. Botev, “Handbook of Monte Carlo Methods”, (Wiley, 2011), Chapter 11.
15	Semester Review
16	Final Exam

Course Notes/Textbooks

David A. Cox, Galois theory, Wiley, 2. Baskı, 2004, ISBN: 9781118072059

Suggested Readings/Materials

Redfield, R. H. Abstract Algebra, A Concrete Introduction, Pearson, 2001.
Stewart, I. Galois Theory, Third edition, Chapman & Hall/CRC, 2003.
Edwards, H. M. Galois Theory, Springer, 1984.
Tignol, J. Galois' theory of algebraic equations, World Scientific, 2001.

Semester Activities	Number	Weigthing
Participation
Laboratory / Application
Field Work
Quizzes / Studio Critiques	1	20
Portfolio	1	10
Homework / Assignments
Presentation / Jury
Project
Seminar / Workshop
Oral Exams
Midterm	1	30
Final Exam		40
Total

Weighting of Semester Activities on the Final Grade	3	60
Weighting of End-of-Semester Activities on the Final Grade	1	40
Total

Semester Activities	Number	Duration (Hours)	Workload
Theoretical Course Hours (Including exam week: 16 x total hours)	16	3	48
Laboratory / Application Hours (Including exam week: '.16.' x total hours)	16		0
Study Hours Out of Class	14	3	42
Field Work			0
Quizzes / Studio Critiques	1	15	15
Portfolio	2	8	16
Homework / Assignments			0
Presentation / Jury			0
Project			0
Seminar / Workshop			0
Oral Exam			0
Midterms	1	28	28
Final Exam		31	0
		Total	149

#	Program Competencies/Outcomes	* Contribution Level
#	Program Competencies/Outcomes	1	2	3	4	5
1	To be able to have a grasp of basic mathematics, applied mathematics or theories and applications of statistics.	-	-	-	-	-
2	To be able to use advanced theoretical and applied knowledge, interpret and evaluate data, define and analyze problems, develop solutions based on research and proofs by using acquired advanced knowledge and skills within the fields of mathematics or statistics.	-	-	X	-	-
3	To be able to apply mathematics or statistics in real life phenomena with interdisciplinary approach and discover their potentials.	-	X	-	-	-
4	To be able to evaluate the knowledge and skills acquired at an advanced level in the field with a critical approach and develop positive attitude towards lifelong learning.	-	-	X	-	-
5	To be able to share the ideas and solution proposals to problems on issues in the field with professionals, non-professionals.	-	-	-	-	-
6	To be able to take responsibility both as a team member or individual in order to solve unexpected complex problems faced within the implementations in the field, planning and managing activities towards the development of subordinates in the framework of a project.	-	-	-	-	-
7	To be able to use informatics and communication technologies with at least a minimum level of European Computer Driving License Advanced Level software knowledge.	-	-	-	-	-
8	To be able to act in accordance with social, scientific, cultural and ethical values on the stages of gathering, implementation and release of the results of data related to the field.	-	-	-	-	-
9	To be able to possess sufficient consciousness about the issues of universality of social rights, social justice, quality, cultural values and also environmental protection, worker's health and security.	-	-	-	-	-
10	To be able to connect concrete events and transfer solutions, collect data, analyze and interpret results using scientific methods and having a way of abstract thinking.	-	-	-	X	-
11	To be able to collect data in the areas of Mathematics or Statistics and communicate with colleagues in a foreign language.	-	-	-	-	-
12	To be able to speak a second foreign language at a medium level of fluency efficiently.	-	-	-	-	-
13	To be able to relate the knowledge accumulated throughout the human history to their field of expertise.	-	-	-	-	-