COURSE UNIT TITLE

: GALOIS THEORY

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 4045 GALOIS THEORY ELECTIVE 4 0 0 7

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics (Evening)
Mathematics

Course Objective

The classical algebra problem is to find a ``formula'' for the roots of polynomial equations; by a formula, we mean a formula 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.

Learning Outcomes of the Course Unit

1   Will be able to formulate the precise meaning of the solvability problem of polynomial equations by radicals.
2   Will be able to distinguish normal and separable 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.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
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. Algebraic extensions. Simple extensions.
6 Splitting fields, their uniqueness up to isomorphism. Normal extensions.
7 Separable extensions. Fields of characteristic 0 and fields of characteristic p. The Primitive Element Theorem.
8 Galois group. Galois group of splitting fields. Permutation of the roots.
9 Midterm
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.

Recomended or Required Reading

Textbook(s): Cox, David A. Galois Theory, Wiley-Interscience, 2004.

Supplementary Book(s):
1) Redfield, R. H. Abstract Algebra, A Concrete Introduction, Pearson, 2001.
2) Stewart, I. Galois Theory, Third edition, Chapman & Hall/CRC, 2003.

References:
1) Edwards, H. M. Galois Theory, Springer, 1984.
2) Rotman, J. Galois theory, Second edition, Springer, 1998.
3) Tignol, J. Galois' theory of algebraic equations, World Scientific, 2001.

Materials: Instructor s notes and presentations

Planned Learning Activities and Teaching Methods

Lecture Notes, Presentation, Problem Solving, Discussion

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE MIDTERM EXAM
2 FIN FINAL EXAM
3 FCG FINAL COURSE GRADE MTE * 0.40 + FIN * 0.60
4 RST RESIT
5 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.40 + FIN * 0.60


*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

e-mail: engin.mermut@deu.edu.tr
Office: (232) 301 85 82

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 13 4 52
Preparation for midterm exam 1 30 30
Preparations before/after weekly lectures 12 4 48
Preparation for final exam 1 30 30
Final 1 3 3
Midterm 1 3 3
TOTAL WORKLOAD (hours) 166

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.153533
LO.253533
LO.353533
LO.453533
LO.553533

CONTACT INFORMATION
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Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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