MATH368 FIELD EXTENSIONS AND GALOIS THEORY
| Course Code: | 2360368 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 6.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Undergraduate |
| Course Coordinator: | Prof.Dr. MUSTAFA HURŞİT ÖNSİPER |
| Offered Semester: | Fall Semesters. |
Course Objectives
The course focuses on fundamentals of field theory, field extensions and Galois theory. In particular, various types of field extensions such as finite, algebraic, transcendental, separable and normal extensions are discussed. A second theme is a brief introduction to Galois theory along with a proof of the fundamental theorem about Galois correspondence. Various applications of Galois theory such as compass-straightedge constructions and solvability by radicals is discussed.
Course Content
Field extensions, splitting field of a polynomial, multiple roots, Galois group, criteria for solvability by radicals, Galois group as permutation groups of the roots of polynomials of degree n, constructible n-gons, transcendence of e, finite fields.
Course Learning Outcomes
At the end of the course a succesful student will
- Identify different types of field extensions and state their main properties,
- Find minimal polynomials, test a given polynomial for irreducibility,
- Compute the set of embeddings of a given field into another,
- Carry out certain impossibility proofs, especially using degree of a field extension,
- Find the splitting field of a set of polynomials,
- Determine if a given element or extension is separable,
- Identify algebraically independent elements and compute transcendence degree,
- Find the lattice of intermediate fields between two fields,
- Compute the Galois group of a field extension,
- Match the lattice of intermediate fields between two fields and subgroups of the Galois group through the Galois correspondence maps.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Can produce innovative thoughts and products. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Has a comprehension of mathematical symbols, concepts together with the interactions among them and can express his/her solutions similarly. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Is responsive to life-long learning, improving his/her skills and abilities | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution