# Galois Theory (dt. Galoistheorie)

 Level, degree of commitment Specialization 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(s): Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written or oral examination (individual examination) Language,Grading English,The grading is done with 0 to 15 points according to the examination regulations for the degree program M.Sc. Mathematics. Duration,frequency One semester, Regularly alternating with other specialization modules in Algebra Person in charge of the module's outline Prof. Dr. Volkmar Welker

## Contents

• Algebraic and transcendental body extensions.
• Decomposition bodies, algebraic closure,
• Normal, separable and inseparable body extensions,
• Galois extensions, main theorem of Galois theory,
• Calculation of the Galois group, translation theorem,
• Finite bodies, unit roots, circular division polynomials,
• Pure equations, cyclic Galois groups,
• Solvability of algebraic equations by radicals (with arbitrary characteristic), constructions with compass and ruler, regular n-corners

## Qualification Goals

Students

• know Galois theory with its applications can evaluate its historical significance in particular,
• understand how elementary problems about geometric constructions and solving equations can be solved by using abstract algebraic methods,
• are trained in the use of algebraic methods by means of many concrete examples,
• have deepened mathematical working methods (developing mathematical intuition and its formal justification, abstraction, proof),
• have improved their oral communication skills in exercises by practicing free speech in front of an audience and in 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.

## Applicability

Module imported from M.Sc. Mathematics.

It can be attended at FB12 in study program(s)

• B.Sc. Mathematics
• M.Sc. Computer Science
• M.Sc. Mathematics
• LAaG Mathematics

When studying LAaG Mathematics, this module can be attended in the study area Advanced Modules.

The module is assigned to Pure Mathematics. Further information on eligibility can be found in the description of the study area.

## Recommended Reading

• Milne, J.S.: Field and Galois Theory, https://www.jmilne.org/math/CourseNotes/ft.html
• Stewart, I.: Galois Theory, London, 2015

## Please note:

This page describes a module according to the latest valid module guide in Winter semester 2023/24. 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.