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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 6023 ADVANCED ALGEBRA ELECTIVE 3 0 0 8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics Mathematics

Course Objective

The aim of this course is to introduce further background topics for research in algebra.

Learning Outcomes of the Course Unit

1   Will be able to understand basic notions for representations of groups. 2   Will be able to understand the basic technique for Lie algebras and Lie groups. 3   Will be able to understand the basic properties of Hopf algebras and algebraic groups. 4   Will be able to use some fundamental theorems of algebraic number theory. 5   Will be able to improve some basic algebraic background for the methods of algebraic geometry.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Groups: Representations of groups, periodic groups, free groups and graphs, representations of groups by generators and relations, simple groups, topological groups. 2 Associative rings: Radical, classical semisimple rings, structure of noetherian rings, central simple algebras, complete rings of fractions. 3 Lie algebras: Linear Lie algebras, universal enveloping algebra, Magnus theory of free groups, Lie algebras with triangular decomposition, Lie algebras and Lie groups. 4 Hopf algebras and algebraic groups, action of an algebraic group, solvable groups. 5 Varieties of Algebras: Universal algebras and varieties, finite basis problems for identities in groups, PI-algebras, central polynomials for matrix algebras. 6 Infinite field extensions: Nullstellensatz, Transcendence degree, separable and purely inseparable extensions. 7 Krull dimension, nonsingular and singular points, infinite Galois groups. 8 Midterm 9 Three important theorem from Algebraic Number Theory: Dedekind discriminant theorem, Dirichlet Unit Theorem and Finiteness of the Class Number. 10 Adeles and Ideles: p-adic numbers, discrete valuations, absolute values, completions, Hensel s Lemma. 11 Ramification indices and residue class degrees, special features of Galois extensions, different and discriminant, global and local fields, adeles and ideles. 12 Backgrounds for Algebraic Geometry: Historical origins and overview, resultant and Bezout s Theorem, projective plane curves, intersection multiplicity for two curves, Bezout s theorem for plane curves, Groebner basis, reduced Groebner basis, systems of polynomial equations. 13 The Number Theory of Algebraic Curves: Divisors, genus, Riemann-Roch Theorem. 14 Methods of Algebraic Geometry: Affine algebraic sets and affine varieties, geometric dimension, projective algebraic sets and projective varieties, rational functions and regular functions, rational points, Zariski s theorem about nonsingular points, Hilbert polynomial.

Recomended or Required Reading

Textbook(s): [1] Bahturin, Y. Basic Structures of Modern Algebra. Kluwer, 1993. [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 ASG ASSIGNMENT 2 MTE MIDTERM EXAM 3 FIN FINAL EXAM 4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40 5 RST RESIT 6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 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

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

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 13 3 39 Preparations before/after weekly lectures 13 5 65 Preparation for midterm exam 1 15 15 Preparation for final exam 1 25 25 Preparing assignments 10 5 50 Final 1 3 3 Midterm 1 3 3 TOTAL WORKLOAD (hours) 200

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 4 4 3 4 1 5 3 LO.2 4 3 3 4 4 3 4 1 5 3 LO.3 4 3 3 4 4 3 4 1 5 3 LO.4 4 3 3 4 4 3 4 1 5 3 LO.5 4 3 3 4 4 3 4 1 5 3