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 | A First Course in Abstract Algebra, Fraleigh |
| Additional Resources | Abstract Algebra, Dummit-Foote |
Material Sharing
| Documents | YULEARN |
| Assignments | |
| Exams |
Assessment
| IN-TERM STUDIES | NUMBER | PERCENTAGE |
| Midterms | 2 | 100 |
| Total | 100 | |
| CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE | 50 | |
| CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE | 50 | |
| Total | 100 |
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 fundamental 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 responsibility | 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 |
| Midterms (Including self study) | 2 | 50 | 50 |
| Final examination (Including self study) | 1 | 26 | 26 |
| Total Work Load | 174 | ||
| Total Work Load / 25 (h) | 6.96 | ||
| ECTS Credit of the Course | 7.00 |