COURSE UNIT TITLE

: ALGEBRAIC NUMBER THEORY

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 4053 ALGEBRAIC NUMBER 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 aim of this course is to introduce the methods of algebraic number theory with the motivation from some classical number theory problems. We shall also see some geometric methods used in the investigations.

Learning Outcomes of the Course Unit

1   Will be able to observe the failure of unique factorization into irreducible elements in some integral domains.
2   Will be able to use factorization of ideals uniquely as a product of prime ideals in Dedekind domains where unique factorisation into irreducibles fail.
3   Will be able to work in quadratic and cyclotomic fields and the ring of algebraic integers of these fields.
4   Will be able to interpret the class-number as a measure of failure of unique factorization into irreducibles.
5   Will be able to use units of the algebraic integers of real quadratic fields for the solution of Pell s equation.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Review of algebraic preliminaries. Review of elementary number theory, primitive roots, quadratic residues, quadratic reciprocity law.
2 Motivation: a) Binary quadratic forms, composition of forms, the class number and the class group, and its relation with quadratic number fields. b) Pell s equation, units in quadratic number fields. c) Fermat s Last Theorem and cyclotomic fields, failure of unique factorization into irreducibles, Fermat s last theorem for regular primes.
3 The ring of algebraic integers. Quadratic and cyclotomic fields.
4 Integral basis. Discriminants. Norms and traces.
5 Factorization into irreducibles. Examples of non-unique factorization into irreducibles.Prime factorization. Euclidean domains. Euclidean quadratic fields. The decomposition of ideals.
6 Dedekind domains. Discrete valuation rings.
7 The norm and classes of ideals. Estimates for the discriminant.
8 Dirichlet's Unit Theorem. Units of quadratic fields and cyclotomic fields.
9 Midterm
10 Extension of ideals. Decomposition of prime numbers in quadratic and cyclotomic fields.
11 The decomposition of prime ideals in Galois extensions.
12 The class group and the finiteness of the class number.Minkowski's Theorems.
13 Valuations, completions, p-adic numbers, Hensel's Lemma.
14 Sketch of some further topics: local and global fields, class field theory, Gaussian sums, zeta functions,elliptic curves, ...

Recomended or Required Reading

Textbook(s):
1) Ribenboim, P. Classical Theory of Algebraic Numbers, Springer, 2001.
2) Alaca, Ş. and Williams, K. S. Introductory Algebraic Number Theory, Cambridge, 2004.
Supplementary Book(s):
1) Stewart, I. Algebraic Number Theory and Fermat s Last Theorem, Third edition, AK Peters, 2002.
2) Milne, J. Algebraic Number Theory (v3.03), 2011, available at www.jmilne.org/math/
References:
1) Edwards, H. M. Fermat s last theorem, A genetic introduction to algebraic number theory, Springer, 1977.
2) Dedekind, R. Theory of algebraic integers, Cambridge, 1996. Translated from the 1877 French original and with an introduction by John Stillwell.
3) Cohn, H. Advanced Number Theory, Dover, 1962.
4) Hasse, H. Number Theory, Springer, 2002 (reprint of the 1980 edition).
5) Ireland, K. and Rosen, M. A Classical Introduction to Modern Number Theory, Second edition, Springer, 1990.
6) Borevich, Z. I. and Shafarevich, I. R. Number Theory, Academic Press, 1966.

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
Preparations before/after weekly lectures 12 4 48
Preparation for midterm exam 1 30 30
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

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