TÜRKÇE
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Information
| Code | MT412 |
| Name | Field Theory |
| Term | 2023-2024 Academic Year |
| Semester | 8. Semester |
| Duration (T+A) | 3-0 (T-A) (17 Week) |
| ECTS | 5 ECTS |
| National Credit | 3 National Credit |
| Teaching Language | Türkçe |
| Level | Lisans Dersi |
| Type | Normal |
| Mode of study | Yüz Yüze Öğretim |
| Catalog Information Coordinator | Prof. Dr. GONCA AYIK |
| Course Instructor |
Prof. Dr. GONCA AYIK
(A Group)
(Ins. in Charge)
|
Course Goal / Objective
The aim of this course is to make students comprehend the Galois theory and some of its results.
Course Content
In this course review of theory of rings, field extension, simple and transcendental extensions, degree of an extension, ruler and compass constructions, Galois group of an extension, splitting fields, normal and separable extensions, solution of equations by radicals are described.
Course Precondition
NONE
Resources
John M. Howie, Fields and Galois theory, Springer- verlag London, 2006
Notes
I. Stewart, Galois Theory, Chapman and Hall, London 1973
Course Learning Outcomes
| Order | Course Learning Outcomes |
|---|---|
| LO01 | Explains the concept of field extension with examples. |
| LO02 | Defines the concept degree of an extension. |
| LO03 | Explains constructable and unconstructable figures using extensions. |
| LO04 | Realises a finite field and can build finite fields. |
| LO05 | Realises splitting field of a polynomial. |
| LO06 | Realises normal and separable extensions and gives examples. |
| LO07 | Recognises the Galois group of an extension and finds the Galois group a given extension. |
| LO08 | Recognises the Galois relation for an extension, decides whether or not the relationshipis is bijection. |
Relation with Program Learning Outcome
| Order | Type | Program Learning Outcomes | Level |
|---|---|---|---|
| PLO01 | Bilgi - Kuramsal, Olgusal | Comprehend the ability to prove the mathematical knowledge gained in secondary education on the basis of theoretical basis. | 5 |
| PLO02 | Bilgi - Kuramsal, Olgusal | Understands importance of basic consepts of Algebra, Analaysis and Topology. | 5 |
| PLO03 | Yetkinlikler - Öğrenme Yetkinliği | Mathematical reasoning demonstrates the ability to develop and write mathematical proofs by gaining maturity. | 4 |
| PLO04 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to express the basic theories of mathematics both correctly. | 3 |
| PLO05 | Bilgi - Kuramsal, Olgusal | Understands the relationship between the different fields of mathematics and its relation to other disciplines. | 5 |
| PLO06 | Bilgi - Kuramsal, Olgusal | Comprehends the ability to understand the relationships between the objects in the most understandable way while creating a model for any problem. | |
| PLO07 | Bilgi - Kuramsal, Olgusal | Comprehend and explain mathematical models such as formulas, graphs, tables and schema. | 5 |
| PLO08 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to mathematically rearrange, analyze, and model the problems they encounter. | 5 |
| PLO09 | Bilgi - Kuramsal, Olgusal | Comprehends at least one of the computer programming languages. | |
| PLO10 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to use scientific methods and appropriate technologies effectively in problem solving. | 4 |
| PLO11 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | Understands sufficient knowledge of foreign language to be able to understand Mathematical concepts and communicate with other mathematicians | |
| PLO12 | Yetkinlikler - Bağımsız Çalışabilme ve Sorumluluk Alabilme Yetkinliği | In addition to their professional development, they demonstrate their ability to continuously improve themselves by identifying their educational needs in scientific, cultural, artistic and social areas in line with their interests and abilities. | 4 |
| PLO13 | Yetkinlikler - Öğrenme Yetkinliği | Understands the programming techniques and shows the ability to do programming. | |
| PLO14 | Yetkinlikler - Öğrenme Yetkinliği | Demonstrates the ability to study mathematics both independently and as a group. | 4 |
| PLO15 | Bilgi - Kuramsal, Olgusal | Demonstrate an awareness of the universal and social impacts and legal consequences of mathematical applications in the field of study. | 5 |
| PLO16 | Bilgi - Kuramsal, Olgusal | Demonstrate the ability to select, use and develop effectively for contemporary mathematical applications. | 4 |
| PLO17 | Bilgi - Kuramsal, Olgusal | It has ability of lifelong learning awareness, access to information, monitoring developments in science and technology and self-renewal ability. | |
| PLO18 | Bilgi - Kuramsal, Olgusal | Gains the ability to use information technologies effectively for contemporary mathematical applications. | |
| PLO19 | Bilgi - Kuramsal, Olgusal | Gains the ability to design, conduct experiments, field work, data collection, analysis, archiving, text solving and / or interpretation according to mathematics fields. | 4 |
| PLO20 | Bilgi - Kuramsal, Olgusal | Gains the consciousness of prefesional ethics and responsibility. | 5 |
Week Plan
| Week | Topic | Preparation | Methods |
|---|---|---|---|
| 1 | Review of basic concepts from ring theory | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 2 | Decomposition in polynomial rings | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 3 | Field extensions | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 4 | Classification of simple extensions | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 5 | Degree of an extension | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 6 | Ruler and compass construction | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 7 | Principles of the Galois Theory | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 8 | Mid-Term Exam | Review of topics discussed in the lecture notes and sources | Ölçme Yöntemleri: Yazılı Sınav |
| 9 | Splitting fields | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 10 | Finite filelds | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 11 | Monomorphisms between fields and Galois groups | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 12 | Normal and separable extensions | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 13 | Normal closure | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 14 | Galois relation | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 15 | Galois relation 1 | Review of related concepts from lecture notes and sources | Öğretim Yöntemleri: Anlatım, Tartışma |
| 16 | Term Exams | Review of topics discussed in the lecture notes and sources | Ölçme Yöntemleri: Yazılı Sınav |
| 17 | Term Exams | Review of topics discussed in the lecture notes and sources | Ölçme Yöntemleri: Yazılı Sınav |
Student Workload - ECTS
| Works | Number | Time (Hour) | Workload (Hour) |
|---|---|---|---|
| Course Related Works | |||
| Class Time (Exam weeks are excluded) | 14 | 3 | 42 |
| Out of Class Study (Preliminary Work, Practice) | 14 | 3 | 42 |
| Assesment Related Works | |||
| Homeworks, Projects, Others | 0 | 0 | 0 |
| Mid-term Exams (Written, Oral, etc.) | 1 | 12 | 12 |
| Final Exam | 1 | 18 | 18 |
| Total Workload (Hour) | 114 | ||
| Total Workload / 25 (h) | 4,56 | ||
| ECTS | 5 ECTS | ||