Course Information
SemesterCourse Unit CodeCourse Unit TitleL+PCreditNumber of ECTS Credits
8MATH456GALOIS THEORY3+036

Course Details
Language of Instruction English
Level of Course Unit First Cycle
Department / Program MATHEMATICS
Mode of Delivery Face to Face
Type of Course Unit Elective
Objectives of the Course The classical algebra problem is to find a ``formula'' for the roots of polynomial equations; by a formula, we mean a rule in terms of the coefficients of the polynomial obtained by just using addition, subtraction, multiplication, division and taking powers or roots (to any degree). Whether this is possible for any polynomial is the question that lead to Galois theory. Starting with this motivating problem, the aim is to develop all the necessary algebraic objects (groups and rings, polynomial rings, fields, field extensions) and their properties whenever needed on the way to solve that classical problem in field theory.
Course Content Cubic and quartic equations. Cardan's Formulas. Symmetric polynomials. Discriminant. Roots of polynomials. The Fundamental Theorem of Algebra. Extension fields. Minimal polynomials. Adjoining elements. Degree of a field extension. Finite extensions. The tower theorem. Algebraic extensions. Simple extensions. Splitting fields, their uniqueness up to isomorphism. Normal extensions. Separable extensions. Fields of characteristic 0 and fields of characteristic p. The Primitive Element Theorem. Galois group. Galois group of splitting fields. Permutation of the roots. Examples of Galois groups. Abelian equations. Galois extensions. The Fundamental Theorem of Galois Theory. Solvability by radicals. Solvable groups. Cyclotomic extensions. Regular polygons and roots of unity. Impossibility of some geometric constructions using just straightedge and compass. Finite fields.
Course Methods and Techniques
Prerequisites and co-requisities None
Course Coordinator None
Name of Lecturers Prof.Dr. Dilek Pusat
Assistants None
Work Placement(s) No

Recommended or Required Reading
Resources Redfield, R. H. Abstract Algebra, A Concrete Introduction, Pearson, 2001.
Stewart, I. Galois Theory, Third edition, Chapman & Hall/CRC, 2003.
Edwards, H. M. Galois Theory, Springer, 1984.
Tignol, J. Galois' theory of algebraic equations, World Scientific, 2001.
Cox, David A. Galois Theory, Wiley-Interscience, 2004.

Course Category

Planned Learning Activities and Teaching Methods
Activities are given in detail in the section of "Assessment Methods and Criteria" and "Workload Calculation"

Assessment Methods and Criteria
In-Term Studies Quantity Percentage
Midterm exams 2 % 40
Quizzes 0 % 0
Homeworks 2 % 20
Other activities 0 % 0
Laboratory works 0 % 0
Projects 0 % 0
Final examination 1 % 40
Total
5
% 100

ECTS Allocated Based on Student Workload
Activities Quantity Duration Total Work Load
Weekly Course Time 3 14 42
Outside Activities About Course (Attendance, Presentation, Midterm exam,Final exam, Quiz etc.) 4 20 80
Exams and Exam Preparations 3 20 60
Total Work Load   Number of ECTS Credits 6 182

Course Learning Outcomes: Upon the successful completion of this course, students will be able to:
NoLearning Outcomes
1 Will be able to formulate the precise meaning of the solvability problem of equations by radicals
2 Will be able to distinguish normal and seperable extensions of fields
3 Will be able to use The Fundamental Theorem of Galois Theory to observe the correspondence between intermediate field extensions and subgroups of the Galois group
4 Will be able to express the solvability of a polynomial equation by radicals using the solvability of its Galois group
5 Will be able to apply Galois Theory to impossibility proofs of some geometric constructions


Weekly Detailed Course Contents
WeekTopicsStudy MaterialsMaterials
1 Cubic and quartic equations. Cardan's Formulas. Classical algebra problem for finding formulae for roots of polynomials in terms of radicals.
2 Symmetric polynomials. Discriminant.
3 Roots of polynomials. The Fundamental Theorem of Algebra.
4 Extension fields. Minimal polynomials. Adjoining elements.
5 Degree of a field extension. Finite extensions. The tower theorem.
6 Algebraic extensions. Simple extensions.
7 Splitting fields, their uniqueness up to isomorphism. Normal extensions.
8 Separable extensions. Fields of characteristic 0 and fields of characteristic p. The Primitive Element Theorem.
9 Galois group. Galois group of splitting fields. Permutation of the roots.
10 Examples of Galois groups. Abelian equations.
11 Galois extensions. The Fundamental Theorem of Galois Theory.
12 Solvability by radicals. Solvable groups.
13 Cyclotomic extensions. Regular polygons and roots of unity. Impossibility of some geometric constructions using just straightedge and compass.
14 Finite fields.


Contribution of Learning Outcomes to Programme Outcomes
P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14
All 3
C1 3 2 4 2 3
C2 3 2 4 2 3
C3 3 2 4 2 3
C4 3 2 4 2 3
C5 3 2 4 2 3

Contribution: 0: Null 1:Slight 2:Moderate 3:Significant 4:Very Significant


https://obs.iyte.edu.tr/oibs/bologna/progCourseDetails.aspx?curCourse=163228&lang=en