312.140 (21W) Algebra

Wintersemester 2021/22

Anmeldefrist abgelaufen.

Erster Termin der LV
06.10.2021 15:30 - 17:00 HS 11 On Campus
... keine weiteren Termine bekannt

Überblick

Bedingt durch die COVID-19-Pandemie können kurzfristige Änderungen bei Lehrveranstaltungen und Prüfungen (z.B. Absage von Präsenz-Lehreveranstaltungen und Umstellung auf Online-Prüfungen) erforderlich sein.

Weitere Informationen zum Lehrbetrieb vor Ort finden Sie unter: https://www.aau.at/corona.
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 06.10.2021
eLearning zum Moodle-Kurs

Zeit und Ort

Beachten Sie bitte, dass sich aufgrund von COVID-19-Maßnahmen die derzeit angezeigten Termine noch ändern können.
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LV-Beschreibung

Intendierte Lernergebnisse

After successful completion of this course, students should know and understand the theory in the areas of algebra listed below. They should be able to apply this knowledge to solving corresponding problems.

Lehrmethodik

Lecture

Inhalt/e

In this course we'll cover some more advanced topics in Group theory, and the introduction to  Galois Theory.  We'll develop some theories and use them to prove several non-trivial theorems. Some of them provide further insight into the general structure of groups and fields; others answer very 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 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 concerning 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 in terms of the coefficients by means of arithmetic operations and taking roots of some degrees. This is the famous Ruffini-Abel Theorem.

6. Another famous application of Galois Theory determines which regular polygons can be constructed by ruler an compass. For example, it is impossible to construct a regular 7-gon and a regular 9-gon, but it is possible to construct a regular 17-gon, a regular 257-gon, and a regular 65537-gon. The facts that these numbers are prime and equal 2^(2^2)+1,  2^(2^3)+1, 2^(2^4)+1, play a crucial role. Gauss, who proved the characterizations of constructable regular polygons, considered this result as one of his most important achievements. According to a legend, he requested to inscribe a regular 17-gon on his gravestone - which was not fulfilled because a regular 17-gon would be hardly distinguishable from a circle.


List of topics:

Part I: Group Theory

  • Group actions.
  • Burnside's Lemma and Pólya's Counting Theory.
  • Sylow's theorems.
  • Structure of finitely generated abelian groups.
  • Simple groups and solvable groups.

Part II: Galois Theory

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

Erwartete Vorkenntnisse

Basic course on abstract algebra (for example that AAU 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. 

We'll begin our course with review / reminder of these topics. The students are asked to read the summary of the material on groups, as was taught in 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:

Israel Herstein. Topics in Algebra.

Nathan Jacobson. Basic Algebra I.

John B. Fraleigh. A First Course in Abstract Algebra.

Joseph Gallian. Contemporary Abstract Algebra.

Also: the lecture notes by Clemens Heuberger. 

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.

Geänderte Prüfungsinformationen (COVID-19 Ausnahmeregelung)

The students are required to be informed about and to follow the up to date COVID regulations. In case of substantial worsening of the epidemic situation, the University might order switching to online teaching (including exams). In such a case students will get all the relevant information in real time. 

Prüfungsmethode/n

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

Prüfungsinhalt/e

Contents of the lecture, typical examples of the learned concepts.

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 Technische Mathematik (SKZ: 401, Version: 13W.1)
    • Fach: Diskrete Mathematik (Pflichtfach)
      • Algebra ( 2.0h VO / 3.0 ECTS)
        • 312.140 Algebra (2.0h VO / 3.0 ECTS)
  • 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 2023/24
  • 312.140 VO Algebra (2.0h / 3.0ECTS)
Wintersemester 2022/23
  • 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)