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German original

Galois Theory
(dt. Galoistheorie)

Level, degree of commitment in original study programme Advanced module, compulsory elective module
Forms of teaching and learning,
workload
Lecture (4 SWS), recitation class (2 SWS),
270 hours (90 h attendance, 180 h private study)
Credit points,
formal requirements
9 CP
Course requirement: Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination
Language,
Grading
German,
The grading is done with 0 to 15 points according to the examination regulations for study course M.Sc. Mathematics.
Original study programme M.Sc. Mathematik / Vertiefungsbereich Mathematik
Duration,
frequency
One semester,
Regelmäßig im Wechsel mit anderen advanced moduleen in Algebra
Person in charge of the module's outline Prof. Dr. István Heckenberger

Contents

  • polynomials in several variables,
  • splitting fields, algebraic closure, Steinitz's theorem,
  • Normal, separable and inseparable field extensions,
  • Galois extensions, fundamental theorem of Galois theory,
  • Computation of the Galois group, translation theorem,
  • Finite fields, roots of unity, cyclotomic polynomials,
  • Pure equations, cyclic Galois groups,
  • solubility of algebraic equations by radicals (in any characteristic), constructions with compass and ruler, regular n-gons

Qualification Goals

The students

1. get to know Galois theory and its applications and, in particular, are able to assess its historical significance,

2. learn how elementary questions about geometric constructions and solving equations can be approached by abstract algebraic methods,

3. train the use of algebraic methods using many concrete examples,

4. practice mathematical methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof methods),

5. improve their oral communication skills in the recitation classes by practicing free speech in front of an audience and during discussion.


Prerequisites

None. The competences taught in the following modules are recommended: either Foundations of Mathematics and Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, Algebra.


Recommended Reading

  • Cigler, J.: Körper, Ringe, Gleichungen, Spektrum.
  • Stewart, I.: Galois Theory, London.



Please note:

This page describes a module according to the latest valid module guide in Sommersemester 2021. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.