MT412 Field Theory

5 ECTS - 3-0 Duration (T+A)- 8. Semester- 3 National Credit

Information

Code MT412
Name Field Theory
Term 2024-2025 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
1 2
Prof. Dr. GONCA AYIK (A Group) (Ins. in Charge)


Course Goal / Objective

The aim of this course is to explain 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 Knows finite fields.
LO05 Finds splitting field of a polynomial.
LO06 Knows normal and separable extensions and gives examples.
LO07 Finds the Galois group of an extension and the Galois group a given extension.
LO08 Finds the Galois relation for an extension and 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. 3
PLO02 Bilgi - Kuramsal, Olgusal Understands importance of basic consepts of Algebra, Analaysis and Topology. 4
PLO03 Yetkinlikler - Öğrenme Yetkinliği Mathematical reasoning demonstrates the ability to develop and write mathematical proofs by gaining maturity. 3
PLO04 Bilgi - Kuramsal, Olgusal Demonstrates the ability to express the basic theories of mathematics accurately both in writing and orally. 3
PLO05 Bilgi - Kuramsal, Olgusal Understands the relationship between the different fields of mathematics and its relation to other disciplines. 3
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. 2
PLO08 Bilgi - Kuramsal, Olgusal Demonstrate the ability to mathematically rearrange, analyze, and model the problems they encounter. 2
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. 3
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. 2
PLO15 Bilgi - Kuramsal, Olgusal Demonstrate an awareness of the universal and social impacts and legal consequences of mathematical applications in the field of study.
PLO16 Bilgi - Kuramsal, Olgusal Demonstrate the ability to select, use and develop effectively for contemporary mathematical applications. 2
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. 3
PLO20 Bilgi - Kuramsal, Olgusal Gains the consciousness of prefesional ethics and responsibility.


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, Soru-Cevap
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, Soru-Cevap
4 Classification of simple extensions Review of related concepts from lecture notes and sources Öğretim Yöntemleri:
Anlatım, Tartışma, Soru-Cevap
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, Alıştırma ve Uygulama
7 Principles of the Galois Theory Review of related concepts from lecture notes and sources Öğretim Yöntemleri:
Anlatım, Tartışma, Soru-Cevap
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, Alıştırma ve Uygulama
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, Soru-Cevap
13 Normal closure Review of related concepts from lecture notes and sources Öğretim Yöntemleri:
Anlatım, Tartışma, Alıştırma ve Uygulama
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, Soru-Cevap
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

Update Time: 17.06.2024 12:16