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German original

Algebraic Equations and Varieties
(dt. Algebraische Gleichungen und Varietäten)

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 und Geometrie
Person in charge of the module's outline Prof. Dr. Thomas Bauer, Prof. Dr. István Heckenberger

Contents

Galois theory: algebraic field extensions, constructions with ruler and compass, normal and separable field extensions, cyclotomic polynomials, finite fields, solvable groups, symmetric polynomials, fundamental theorem of Galois theory, solvability of algebraic equations

Algebraic varieties: Affine varieties and Hilbert Nullstellensatz, Morphisms of Affine Varieties, Rational Functions and maps, Smooth Points, Tangent Spaces and Dimensions


Qualification Goals

The students shall

  • learn the essential features of Galois theory and its applications and appreciate its historical significance,
  • learn about the applicability of algebraic methods for the description of geometric objects (algebraic varieties),
  • Understand the translation process between geometry and algebra and be able to apply it to posed problems,
  • practice mathematical working methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
  • improve their oral communication skills in the exercises by practicing free speech in front of an audience and during discussion.

Prerequisites

None. The competences taught in the following modules are recommended: either Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, either Elementary Algebraic Geometry or Algebra.


Recommended Reading

  • G. Fischer, R. Sacher: Einführung in die Algebra. Teubner.
  • K. Hulek: Elementare Algebraische Geometrie. Vieweg.



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.