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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 5048 ALGEBRA II 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 fields and Galois theory and some basic results from the following topics: algebraic number theory, cohomology of groups, categories and functors, and homological algebra.

Learning Outcomes of the Course Unit

1   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. 2   Will be able to express the solvability of a polynomial equation by radicals using the solvability of its Galois group. 3   Will be able to apply Galois Theory to impossibility or constructability proofs of some geometric constructions. 4   Will be able to understand the motivation for some important concepts in commutative algebra and field theory that comes from algebraic number theory. 5   Will be able to use the projective and injective resolutions to obtain the derived functors Ext and Tor of the functors Hom and tensor product.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Motivation for Commutative Algebra: Algebraic number theory and algebraic geometry. 2 Noetherian rings, Hilbert Basis Theorem, Integral Closure. 3 Localization and local rings, Dedekind domains. 4 Fields and Galois Theory: Algebraic elements, field extensions, finite fields, algebraic closure, geometric constructions by straightedge and compass. 5 Separable extensions, normal extensions, Fundamental Theorem of Galois Theory, solvable groups. 6 Applications of Galois Theory: Constructibility of regular polygons, Proof of Fundamental Theorem of Algebra, Unsolvability of polynomial equations with nonsolvable Galois group. 7 Norm and trace, splitting of prime ideals in extensions, computing Galois groups. 8 Midterm 9 Transition to Modern Number Theory: Historical background, Quadratic reciprocity, equivalence and reduction of quadratic forms, composition of forms, class group, genera. 10 Quadratic number fields and their units, relationship of quadratic forms to ideals, Dirichlet s series and Euler products, Dirichlet s Theorem on primes in arithmetic progressions. 11 Brauer group: Factor sets, crossed products, Hilbert s Theorem 90, cohomology of groups. 12 Categories and functors. 13 Homological Algebra: Complexes and additive funtors, Projectives and Injectives. 14 Long exact sequences of derived functors, Ext and Tor.

Recomended or Required Reading

Textbook(s): [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 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

Further Notes About Assessment Methods

None

Assessment Criteria

50% success rate on each of the assessments.

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 of the course at the beginning of each semester.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 14 3 42 Preparation for midterm exam 14 3 42 Preparations before/after weekly lectures 1 15 15 Preparation for final exam 1 20 20 Preparing assignments 10 5 50 Final 1 3 3 Midterm 1 2 2 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