312.262 (17S) Selected Topics in Algebra and Number Theory

Sommersemester 2017

Registration deadline has expired.

First course session
02.03.2017 10:00 - 12:00 HS 2 On Campus
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Overview

Lecturer
Course title german Ausgewählte Kapitel der Algebra und Zahlentheorie
Type Lecture - Practical class (continuous assessment course )
Hours per Week 3.0
ECTS credits 5.0
Registrations 4 (25 max.)
Organisational unit
Language of instruction English
possible language(s) of the assessment German , English , French
Course begins on 02.03.2017
eLearning Go to Moodle course
Remarks (english)

In case of scheduling conflicts, please contact the lecturer ... we will try to find a time slot suitable for everybody.

Prerequisites: basic course in Algebra ("Algebraische Strukturen" is sufficient) and number theory.

Seniorstudium Liberale Yes

Time and place

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Course Information

Intended learning outcomes

After successful completion of this course, students know the definitions and theorems in algebraic number theory as listed in the contents section below. They understand and are able to prove these theorems and are able to apply these results to solve pertinent problems.

Teaching methodology including the use of eLearning tools

Lecture. Solving assignments and presenting the results.

Course content

Introduction to Algebraic Number Theory.

We study "algebraic integers", i.e., algebraic numbers whose minimal polynomials have integer coefficients. The ring of integers of a number field (finite extension of the rational numbers) is the analogon of the integers in the field of rationals.

After a general introduction, the following classical results are studied:

  • Dirichlet's unit theorem. In general, the ring of integers of a number field has many units. Dirichlet's unit theorem gives a precise description of the unit group. This generalises the theory of Pell's equation.
  • Dedekind rings and ideal decomposition. In general, a ring of integers in a number field is no longer a factorial ring. However, there is unique factorisation of ideals into prime ideals and this factorisation is a very useful tool.
  • Finiteness of the class group. Obviously, a ring of integers cannot be a principal ideal in general. The so-called class group measures the deviation of a ring of integers from a principal ideal. Its cardinality, the class number, is shown to be finite in all cases.

Literature

Lang, Algebraic Number Theory.

Borevich-Shafarevich, Number Theory.

Examination information

Im Fall von online durchgeführten Prüfungen sind die Standards zu beachten, die die technischen Geräte der Studierenden erfüllen müssen, um an diesen Prüfungen teilnehmen zu können.

Examination methodology

Oral exam plus solving and presenting assignment problems.

Examination topic(s)

Contents of the lecture and assignments.

Assessment criteria / Standards of assessment for examinations

The grades for lecture and practical are weighted by 75% and 25%, respectively.

The grade for the lecture is obtained by oral exam.

For the practical, 16 points can be achieved. There will be 5 practical sessions of 75 minutes; the best four sessions are counted. Students are expected to present the assignments handed out via Moodle a week before the practical session. Per practical session, the quantity and quality of presentations of each student will be rewarded freely with up to 4 points.

Grading scheme

Grade / Grade grading scheme

Position in the curriculum

  • Master's degree programme Technical Mathematics (SKZ: 401, Version: 13W.1)
    • Subject: Diskrete Mathematik (Compulsory elective)
      • Ausgewählte Kapitel der Algebra und Zahlentheorie ( 3.0h VU / 5.0 ECTS)
        • 312.262 Selected Topics in Algebra and Number Theory (3.0h VU / 5.0 ECTS)

Equivalent courses for counting the examination attempts

This course is not assigned to a sequence of equivalent courses