312.262 (17S) Ausgewählte Kapitel der Algebra und Zahlentheorie

Sommersemester 2017

Anmeldefrist abgelaufen.

Erster Termin der LV
02.03.2017 10:00 - 12:00 , HS 2
... keine weiteren Termine bekannt

Überblick

Lehrende/r
LV-Titel englisch
Selected Topics in Algebra and Number Theory
LV-Art
Vorlesung-Übung (prüfungsimmanente LV )
Semesterstunde/n
3.0
ECTS-Anrechungspunkte
5.0
Anmeldungen
4 (25 max.)
Organisationseinheit
Unterrichtssprache
Englisch
mögliche Sprache/n der Leistungserbringung
Deutsch , Englisch , Französisch
LV-Beginn
01.03.2017
eLearning
zum Moodle-Kurs
Anmerkungen

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
Ja

LV-Beschreibung

Intendierte Lernergebnisse

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.

Lehrmethodik

Lecture. Solving assignments and presenting the results.

Inhalt/e

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.

Literatur

Lang, Algebraic Number Theory.

Borevich-Shafarevich, Number Theory.

Prüfungsinformationen

Prüfungsmethode/n

Oral exam plus solving and presenting assignment problems.

Prüfungsinhalt/e

Contents of the lecture and assignments.

Beurteilungskriterien/-maßstäbe

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.

Beurteilungsschema

Note/Grade Benotungsschema

Position im Curriculum

  • Masterstudium Technische Mathematik (SKZ: 401, Version: 13W.1)
    • Fach: Diskrete Mathematik (Wahlfach)
      • Ausgewählte Kapitel der Algebra und Zahlentheorie ( 3.0h VU / 5.0 ECTS)
        • 312.262 Ausgewählte Kapitel der Algebra und Zahlentheorie (3.0h VU / 5.0 ECTS)

Gleichwertige Lehrveranstaltungen im Sinne der Prüfungsantrittszählung

Diese Lehrveranstaltung ist keiner Kette zugeordnet