## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Course MAT-40-34-K-4

## Course Type

SWS Type Course Form CP (Effort) Presence-Time / Self-Study
- K Lecture with exercise classes (V/U)
4 V Lecture 6.0 CP 56 h 124 h
2 U Exercise class (in small groups) 3.0 CP 28 h 62 h
(4V+2U) 9.0 CP 84 h 186 h

## Basedata

SWS 4V+2U 9.0 CP = 270 h 1 Sem. irreg. SuSe [4] Bachelor (Specialization) [EN] English Fieker, Claus, Prof. Dr. (PROF | DEPT: MAT) Malle, Gunter, Prof. Dr. (PROF | DEPT: MAT) Thiel, Ulrich, Prof. Dr. (PROF | DEPT: MAT) + further Lecturers of the department Mathematics [MAT-AGCA] Algebra, Geometry and Computer Algebra [NORM] Active

## Notice

The course consists of the two parts [MAT-40-25-K-4] and [MAT-40-24-K-4].

## Possible Study achievement

• Verification of study performance: proof of successful participation in the exercise classes (ungraded)
The certificate for the exercises ("Übungsschein") consists of the two (partial) certificates for the exercises in the courses [MAT-40-25-K-4] and [MAT-40-24-K-4].

## Contents

Character Theory of Finite Groups:

• Maschke's theorem,
• character tables,
• orthogonality relations,
• rationality questions,
• Burnside theorem,
• induced characters,
• Frobenius groups.

• Hensel's lemma,
• algebraic closure,
• Newton polygon,
• inertia and ramification groups

## Competencies / intended learning achievements

In the part "Character Theory of Finite Groups", the students have studied the basic terms, propositions and techniques of representation theory through the character theory of finite groups. In particular, they have learnt to work with generic characters and character tables.

In the part "p-adic Numbers", the students have studied extensions of a number system through p-adic numbers which are fundamental for number theory. In particular, they have studied their main properties and some simple applications.

## Literature

Character Theory of Finite Groups:

• M. Isaacs: Character Theory of Finite Groups,
• G. James, M. Liebeck: Representations and Characters of Finite Groups,
• J. Alperin, R. Bell: Groups and Representations.

• I.N. Stewart, D.O. Tall: Algebraic Number Theory,
• M. Trifković: Algebraic Theory of Quadratic Numbers.

## Materials

Further literature will be announced in the lecture; Exercise material is provided.

## Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de)

## Requirements for attendance (informal)

Knowledge from the course [MAT-12-21-K-3] is useful but not necessarily required.

None

## References to Course [MAT-40-34-K-4]

Course-Pool Name
[MAT-40-4V-KPOOL-4] Elective Courses Algebra, Geometry and Computeralgebra (4V, B.Sc.)
[MAT-40-KPOOL-4] Specialisation Algebra, Geometry and Computer Algebra (B.Sc.)