312.140 (23W) Algebra

Wintersemester 2023/24

Anmeldefrist abgelaufen.

Erster Termin der LV
10.10.2023 12:00 - 14:00 N.2.01 On Campus
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Überblick

Lehrende/r
LV-Titel englisch Algebra
LV-Art Vorlesung
LV-Modell Präsenzlehrveranstaltung
Semesterstunde/n 2.0
ECTS-Anrechnungspunkte 3.0
Anmeldungen 14
Organisationseinheit
Unterrichtssprache Englisch
LV-Beginn 10.10.2023
eLearning zum Moodle-Kurs

Zeit und Ort

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LV-Beschreibung

Intendierte Lernergebnisse

At this course, the students should gain the knowledge of the topics listed below. They should understand definitions, results, proofs, and examples, and should be able to apply basic methods for solving problems.

Lehrmethodik

Lecture

Inhalt/e

In this course we will cover advanced topics in Group theory, and the introduction to  Galois Theory.  We will develop some theory and apply it to prove several non-trivial theorems. Some of the results provide further insight into the general structure of groups and fields; others answer natural questions from other areas of mathematics, such as combinatorics and calculus. An emphasis will be put on intuitive understanding / visualisation of the results. The "highlights" of the course include:

1. Burnside's lemma, which can be used for counting geometric objects "considered up to a symmetry". (Example: In how many ways can one colour the faces of a cube, using three colours, if the colourings obtained from each other by rotating the cube, are considered identical?)

2. A complete classification of finite abelian groups. (With this theorem, it is immediate to give the complete list of all abelian groups, up to isomorphism, of any given finite order.)

3. Sylow's Theorems can be regarded as a partial converse to Lagrange's Theorem.  For many values of n, these theorems can be used for complete classification of all groups of order n. Moreover, Sylow's Theorems can be used for checking whether a group of given order has non-trivial normal groups: this has important implications in Galois theory.

4. Galois Theory makes it possible to reformulate certain questions on fields in terms of finite groups, which allows their better understanding and easier solution. The following two items are classical applications of Galois Theory.

5. The formula for solving quadratic equations is known from the ancient times; formulas for solving polynomial equations  of degrees 3 and 4 were developed in the 16th century. However, no formula is possible for equations of higher degree, and "generically" the roots of such an equation cannot be expressed via its coefficients by means of arithmetic operations and taking roots. This is the famous Ruffini-Abel Theorem.

6. Another famous application of Galois Theory determines which regular polygons can be constructed by a ruler and a compass. For example, it is not possible to construct a regular 7-gon, but it is possible to construct a regular 17-gon, a regular 257-gon, and a regular 65537-gon. Carl Friedrich Gauss, who proved the characterizations of constructable regular polygons, considered this result as one of his most important mathematical achievements. According to a legend, he requested to inscribe a regular 17-gon on his gravestone - which was not done because a regular 17-gon would be hardly distinguishable from a circle.


List of topics:

Part I: Group Theory

  • Group actions. Orbits and stabilizers.
  • Burnside's Lemma and Pólya's Counting Theory.
  • Sylow's theorems.
  • The Fundamental Theorem of finitely generated abelian groups.
  • Simple groups and solvable groups, Jordan-Hölder Theorem.

Part II: Galois Theory

  • Field extensions and splitting fields.
  • Galois theory.
  • Cyclotomic fields.
  • Radical extensions.
  • Solvability of algebraic equations.
  • Geometric constuctions.

Erwartete Vorkenntnisse

A basic course on abstract algebra (for example the BA-course "Algebraische Strukturen"): Groups, subgroups, cyclic groups, Lagrange's Theorem, normal subgroups, quotient group, group homomorphism and isomorphism, Fundamental Theorem of group homomorphism, direct product, symmetric and dihedral groups; rings, ideals, polynomial rings, Euclidean algorithm; fields, simple algebraic field extensions. 

The students are required to read the summary of the material on groups taught at the course Algebraische Strukturen, before the first lecture. This summary can be found at the Moodle page.

Curriculare Anmeldevoraussetzungen

n/a

Literatur

Any introductory book on Group Theory and Galois Theory, for example:

David Dummit and Richard Foote. Abstract Algebra.
Nathan Jacobson. Basic Algebra I.
Israel Herstein. Topics in Algebra.
John B. Fraleigh. A First Course in Abstract Algebra.
Nathan Carter. Visual Group Theory.
David Cox. Galois Theory.

Also: The lecture notes by Clemens Heuberger. 

NB: The course won't be completely based on either of these sources.

Link auf weitere Informationen

https://www.google.com/

Prüfungsinformationen

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.

Prüfungsmethode/n

Oral exam (45-60 minutes). The dates will be fixed with students individually.

Prüfungsinhalt/e

Contents of the lecture: Definitions, main results and their proofs, standard examples.

Beurteilungskriterien/-maßstäbe

Emphasis is laid on reasonable knowledge of the definitions and results, and being able to illustrate them by means of standard examples. 

Beurteilungsschema

Note Benotungsschema

Position im Curriculum

  • Doktoratsprogramm Modeling-Analysis-Optimization of discrete, continuous and stochastic systems (SKZ: ---, Version: 16W.1)
    • Fach: Modeling-Analysis-Optimization of discrete, continuous and stochastic systems (Pflichtfach)
      • Modeling-Analysis - Optimization of discrete, continuous and stochastic systems ( 0.0h XX / 0.0 ECTS)
        • 312.140 Algebra (2.0h VO / 3.0 ECTS)
  • Masterstudium Mathematics (SKZ: 401, Version: 18W.1)
    • Fach: Discrete Mathematics (Pflichtfach)
      • 2.1 Algebra ( 2.0h VO / 3.0 ECTS)
        • 312.140 Algebra (2.0h VO / 3.0 ECTS)
  • Masterstudium Mathematics (SKZ: 401, Version: 22W.1)
    • Fach: Discrete Mathematics (Pflichtfach)
      • 2.1 Algebra ( 2.0h VO / 3.0 ECTS)
        • 312.140 Algebra (2.0h VO / 3.0 ECTS)
          Absolvierung im 1. Semester empfohlen
  • Doktoratsstudium Doktoratsstudium der Technischen Wissenschaften (SKZ: 786, Version: 12W.4)
    • Fach: Studienleistungen gem. § 3 Abs. 2a des Curriculums (Pflichtfach)
      • Studienleistungen gem. § 3 Abs. 2a des Curriculums ( 16.0h XX / 32.0 ECTS)
        • 312.140 Algebra (2.0h VO / 3.0 ECTS)

Gleichwertige Lehrveranstaltungen im Sinne der Prüfungsantrittszählung

Wintersemester 2022/23
  • 312.140 VO Algebra (2.0h / 3.0ECTS)
Wintersemester 2021/22
  • 312.140 VO Algebra (2.0h / 3.0ECTS)
Wintersemester 2020/21
  • 312.140 VO Algebra (2.0h / 3.0ECTS)
Wintersemester 2019/20
  • 312.140 VO Algebra (2.0h / 3.0ECTS)
Wintersemester 2018/19
  • 312.140 VO Algebra (2.0h / 3.0ECTS)
Wintersemester 2017/18
  • 312.140 VO Algebra (2.0h / 3.0ECTS)
Wintersemester 2016/17
  • 312.140 VO Algebra (2.0h / 3.0ECTS)
Wintersemester 2015/16
  • 312.140 VO Algebra (2.0h / 3.0ECTS)
Wintersemester 2014/15
  • 312.140 VO Algebra (2.0h / 3.0ECTS)