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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 5007 ALGEBRA I ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

PROFESSOR DOCTOR NOYAN FEVZI ER

Offered to

Mathematics (English) Mathematics (English)

Course Objective

The aim of this course is to introduce the fundamental concepts in linear and multilinear algebra, group theory, ring theory and module theory.

Learning Outcomes of the Course Unit

1   Will be able to use the basic results in linear and multilinear algebra. 2   Will be able to understand the basic notions and theorems of group theory, and the basic matrix groups. 3   Will be able to use the properties of commutative rings and modules over commutative rings. 4   Will be able to use the structure of finitely generated modules over a PID to obtain the structure of finitely generated abelian groups and canonical forms of linear transformations. 5   Will be able to understand the structure of semisimple rings determined by the Wedderburn-Artin Theorem.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Review of Basic Linear Algebra: Vector spaces, linear maps, dual spaces, quotient spaces, direct sums and products, inner product spaces, adjoint, spectral theorem, determinants of matrices with entries in a commutative ring. 2 Theory of a Single Linear Transformation: Eigenvectors, characteristic and minimal polynomials, primary decomposition, Jordan canonical form, Rational Canonical Form. 3 Multilinear Algebra: Bilinear forms, symmetric bilinear forms, quadratic forms, Hermitian forms, the tensor prouct of vector spaces, tensor algebra, symmetric algebra, exterior algebra. 4 Group Theory: Groups, subgroups, homomorphisms, quotient groups, group actions, semidirect products, simple groups and composition seris, structure of finitely generated abelian groups, Sylow theorems. 5 Free groups, subgroups of free groups, free products of groups. 6 Group representations, Burnside s Theorem. 7 Extensions of groups. 8 Midterm 9 Commutative Rings and Modules over Commutative Rings: integral domains and fields of fractions, prime and maximal ideals, unique factorization, Gauss s Lemma. 10 Finitely generated modules over a principal ideal domain and its application to finitely generated abelian groups and to canonical forms of linear transformations. 11 Modules over Noncommutative Rings: simple and semisimle modules, composition series, chain conditions, Hom and End for modules. 12 Tensor product of modules over noncommutative rings, exact sequences. 13 Wedderburn-Artin Ring Theory: Semisimple rings and Wedderburn s Theorem, rings with chain conditions and Artin s theorem, Wedderburn-Artin radical, semisimplicity and tensor products. 14 Skolem-Noether theorem, Double centralizer theorem, Wedderburn s Theorem about finite division rings, Frobenius s Theorem about division algebras over the real numbers.

Recomended or Required Reading

[1] Knapp, A. W. Basic Algebra, Birkhauser, 2006. [2] Knapp, A. W. Advanced Algebra, Birkhauser, 2007.

Planned Learning Activities and Teaching Methods

Lecture Notes, Presentation, Problem Solving

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 MTE 1 * 0.30 + MTE 2 * 0.30 + FIN * 0.40 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) MTE 1 * 0.30 + MTE 2 * 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

50% success rate on each of the assessment methods applied.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

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

Office Hours

As announced by the instructor during the first week of classes of each semester.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 14 3 42 Preparations before/after weekly lectures 14 2 28 Preparation for midterm exam 2 10 20 Preparation for final exam 1 27 27 Preparing assignments 10 5 50 Final 1 3 3 Midterm 2 2 4 TOTAL WORKLOAD (hours) 174

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 LO.1 4 3 3 2 2 2 4 3 3 3 LO.2 4 3 3 2 2 2 4 3 3 3 LO.3 4 3 3 2 2 2 4 3 3 3 LO.4 4 3 3 2 2 2 4 3 3 3 LO.5 4 3 3 2 2 2 4 3 3 3