COURSE UNIT TITLE

: ALGEBRA II

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 3046 ALGEBRA II 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 aim of this course (and the course MAT3055 Algebra I) is to learn the basic concepts of algebra, the classical topics: Groups, Rings and Fields. In the course MAT3055 Algebra I, we have concentrated on groups. Now we shall study rings and fields in this course MAT3046 Algebra II. The main aim of this course is to study Galois Theory of Fields. We shall start with the classical algebra problem: finding a formula for the roots of polynomial equations. This question leads to Galois theory. We shall follow this historical motivation to understand how algebra evolved as it is today. We shall continue to study some more topics about groups, rings and fields that are covered in the course Basic Algebraic Structures .

Learning Outcomes of the Course Unit

1   Rings, homomorphisms and ideals, quotient rings with the basic examples should be known.
2   Integral domains, Euclidean domains, principal ideal domains, unique factorization domains and Noetherian rings should be known.
3   Field extensions and adjoining algebraic elements to a field should be known.
4   Finite and algebraic field extensions, splitting fields and automorphisms of fields should be known.
5   The Galois correspondence and the fundamental theorem of Galois Theory should be known.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Motivating questions for algebra, a brief historical introduction to Galois theory. Solving the Cubic Equation. The classical algebra problem: finding a formula for the roots of polynomial equations. This leads to Galois theory.
2 Rings. Homomorphisms and Ideals. Quotient Rings.
3 Integral Domains. Euclidean Domains, Principal Ideal Domains, and Unique Factorization.
4 Unique Factorization Domains.
5 Noetherian Rings. Irreducibility Criteria.
6 Field Extensions. Adjoining Algebraic Elements to a Field.
7 Splitting Field of a Cubic Polynomial. Splitting Fields of Polynomials in C[x].
8 Midterm
9 Finite and Algebraic Extensions. Splitting Fields. The Derivative and Multiple Roots. Splitting Fields and Automorphisms.
10 The Galois Correspondence. Symmetric Functions. The General Equation of Degree n.
11 Quartic Polynomials. Galois Groups of Higher Degree Polynomials.
12 Solvability. Composition Series and Solvable Groups. Commutators and Solvability.
13 Simplicity of the Alternating Groups. Cyclotomic Polynomials.
14 Solvability by Radicals. Radical Extensions.

Recomended or Required Reading

Textbook:

Frederick M. Goodman. Algebra, Abstract and Concrete, Stressing Symmetry. Pearson, 2003. Online edition 2.6: http://homepage.divms.uiowa.edu/~goodman/algebrabook.dir/download.htm


Supplementary textbooks:

[1] John B. Fraleigh. A First Course in Abstract Algebra.
Seventh edition. Pearson, 2003.

[2] William J. Gilbert and W. Keith Nicholson. Modern Algebra with Applications. Second edition. John Wiley & Sons, 2004.

[3] Joseph A. Gallian. Contemporary Abstract Algebra. Ninth edition. Cengage Learning, 2017.

[4] Michael Artin. Algebra. Second edition, Pearson, 2010.

[5] Joseph J. Rotman. A First Course in Abstract Algebra with Applications. Third edition, Pearson, 2006.

[6] David S. Dummit and Richard M. Foote. Abstract Algebra. Third edition. John Wiley & Sons, 2004.

[7] M. A. Armstrong. Groups and Symmetry. Springer, 1988.

[8] Nathan C. Carter. Visual Group Theory Mathematical Association of America, 2009.

[9] David W. Farmer. Groups and Symmetry, A Guide to Discovering Mathematics. AMS, 1996.

[10] Elbert A. Walker. Introduction to Abstract Algebra. Random House/Birkhauser, 1987. Online available:
http://emmy.nmsu.edu/~elbert/

[11] John Stillwell. Elements of Algebra. Springer, 1994.

[12] Robert H. Redfield. Abstract Algebra, A Concrete Introduction. Pearson, 2001.

[13] Israel Kleiner. A History of Abstract Algebra. Birkha user, 2007.

[14] Halil I brahim Karakas . Cebir Dersleri. TU BA Ders Kitapları Dizisi Sayı 4, 2008.

Planned Learning Activities and Teaching Methods

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE 1 MIDTERM EXAM 1
2 MTE 2 MIDTERM EXAM 2
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE MTE1 * 0.30 + MTE2 * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) MTE1 * 0.30 + MTE2 * 0.30 + RST * 0.40


*** 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
Phone: (232) 30 18582

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 13 4 52
Preparations before/after weekly lectures 12 4 48
Preparation for midterm exam 2 17 34
Preparation for final exam 1 35 35
Final 1 2 2
Midterm 2 2 4
TOTAL WORKLOAD (hours) 175

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.13453434
LO.25543434
LO.34543434
LO.44543434
LO.54543434

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

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