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 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