GALOIS THEORY

Course Code:

MATH 422

Course Type:

Area Elective

P:

3

Lab:

0

Laboratuvar Saati:

0

Credits:

3

ECTS:

7

Course Language:

English

Course Objectives:

To introduce the basic facts about field extensions, Galois theory and its applications.

Course Content:

Algebraic extensions, Algebraic Closure, Splitting Fields, Normal Extensions, Separable Extensions, Finite Fields, Fundamental Theorem of Galois Theory, Cyclic Extensions, Solvability by Radicals, Solvability of Algebraic Equations, Construction with Ruler and Compass.

Course Methodology:

1: Lecture, 2: Problem Solving

Course Evaluation Methods:

A: Written examination, B: Homework

Vertical Tabs

Course Learning Outcomes

Learning Outcomes	Teaching Methods	Assessment Methods
1) Applies irreducibility criterions to decide if a given polynomial is irreducible or not. Computes the minimal polynomial of a given element algebraic over a base field.	1	A,B
2) Computes the splitting field of a given polynomial	1	A,B
3) Decides if a given polynomial is separable or not.	1	A,B
4) Decides if a given extension is Galois or not. Compute the Galois group of a given Galois extension.	1	A,B
5) Applies Fundamental Theorem of Galois Theory in concrete examples	1	A,B
6) Analyses particular polynomials – computes their Galois groups and assesses their solvability by radicals.	1	A,B

Course Flow

Week	Topics	Study Materials
1	Rings and homomorphisms Ideals and quotient rings	Textbook
2	Polynomial rings Vector spaces	Textbook
3	Algebraic extensions	Textbook
4	Algebraic extensions continued	Textbook
5	Algebraic Closure	Textbook
6	Splitting Fields, Normal Extensions	Textbook
7	Separable Extensions	Textbook
8	Finite Fields	Textbook
9	Fundamental Theorem of Galois Theory	Textbook
10	Fundamental Theorem of Galois Theory continued	Textbook
11	Cyclic Extensions	Textbook
12	Solvability by Radicals	Textbook
13	Solvability of Algebraic Equations	Textbook
14	Construction with Ruler and Compass	Textbook

Recommended Sources

Textbook	Galois Theory, M. P. Murthy, K.G. Ramanathan, C.S. Seshadri, U. Shukla, R. Sridharan, Tata Inst. of Fund. Research, Bombay, 1965
Additional Resources	Algebra, Serge Lang, 3rd. ed., Addison-Wesley, 1994

Material Sharing

Documents
Assignments
Exams

Assessment

IN-TERM STUDIES	NUMBER	PERCENTAGE
Mid-terms
Quizzes
Assignments	7	100
Total		100
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE		40
CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE		60
Total		100

COURSE CATEGORY

Expertise/ Field Courses

Course’s Contribution to Program

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry)			x
2	Acquiring fundamental knowledge on fundamental research fields in mathematics					x
3	Ability form and interpret the relations between research topics in mathematics					x
4	Ability to define, formulate and solve mathematical problems					x
5	Consciousness of professional ethics and responsibilty					x
6	Ability to communicate actively			x
7	Ability of self-development in fields of interest					x
8	Ability to learn, choose and use necessary information technologies		x
9	Lifelong education					x

ECTS

Activities	Quantity	Duration (Hour)	Total Workload (Hour)
Course Duration (14x Total course hours)	14	3	42
Hours for off-the-classroom study (Pre-study, practice)	14	4	56
Mid-terms (Including self study)
Quizzes
Assignements	7	8	56
Final examination (Including self study)	1	20	20
Total Work Load			174
Total Work Load / 25 (h)			6.96
ECTS Credit of the Course			7.00

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