## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Module MAT-02-11-M-1

## Module Identification

Module Number Module Name CP (Effort)
MAT-02-11-M-1 Mathematics for Computer Science Students: Algebraic Structures 8.0 CP (240 h)

## Basedata

CP, Effort 8.0 CP = 240 h 1 Sem. in WiSe/SuSe  Bachelor (General) [DE] German Schweitzer, Pascal, Prof. Dr. (PROF | DEPT: INF) Schulze, Mathias, Prof. Dr. (PROF | DEPT: MAT) Böhm, Janko, Dr. (WMA | DEPT: MAT) Kunte, Michael, Dr. (WMA | DEPT: MAT) [MAT-Service] Mathematics for other Departments [INF-82.79-SG] B.Sc. Computer Science [NORM] Active

## Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
4V+2U MAT-02-11-K-1
Mathematics for Computer Science Students: Algebraic Structures
P 84 h 156 h
U-Schein
ja PL1 8.0 WiSe/SuSe
• About [MAT-02-11-K-1]: Title: "Mathematics for Computer Science Students: Algebraic Structures"; Presence-Time: 84 h; Self-Study: 156 h
• About [MAT-02-11-K-1]: The study achievement "[U-Schein] proof of successful participation in the exercise classes (ungraded)" must be obtained.
• It is a prerequisite for the examination for PL1.

## Examination achievement PL1

• Form of examination: written exam (Klausur) (120-150 Min.)
• Examination number: 81041 ("Algebraic Structures")

## Contents

• propositions, sets, methods of proof, mappings, semi-orderings and equivalence relations,
• integers, division with remainder, greatest common divisor and Euclidean algorithm, fundamental theorem of arithmetic, Chinese remainder theorem over ℤ,
• groups, orbit equation, symmetry groups, normal subgroup and quotient group, applications (e.g. counting isomorphism classes of graphs),
• rings, polynomial rings, unity group of ℤ/nℤ, applications (e.g. public key cryptography, Pollard factorization, Diffie-Hellman key exchange), ideals and quotient rings, integral domains and fields, finite fields, Euclidean rings, Chinese remainder theorem, applications (e.g. modular computing, interpolation),
• vector spaces, Gaussian algorithm, bases and dimension, vector space homomorphisms, solving linear systems of equations, representing matrix of a homomorphism, algorithms for kernel and image,
• isomorphisms, base change, application (e.g. wavelet transformation), classification of homomorphisms, homomorphism theorem, applications (e.g. linear codes), determinants, eigenvectors, applications (e.g. page-rank algorithm).

## Competencies / intended learning achievements

Upon successful completion of this module, the students have achieved the following learning outcomes:
• The students know the fundamental formation of mathematical concepts and basic proof methods in mathematics (in particular, proof by contraposition and proof by induction).
• On the basis of a structure-oriented approach, they are trained in analytical thinking and their ability to think abstractly has been promoted.
• They have learned to comprehend mathematical proofs and are able to prove or disprove mathematical statements independently in simple examples.
• They know and understand the basic concepts of elementary number theory, algebra and linear algebra. Through an algorithmic approach they have learned the fundamental statements and methods of these fields and are able to apply them to problems in computer science.
• In the exercise classes they have acquired a confident, precise and independent handling of the terms, statements and methods from the lecture.
• In addition, their presentation skills and ability to work in a team have been promoted.

## Literature

• G. Fischer: Lineare Algebra,
• S. Bosch: Lineare Algebra,
• K. Jänich: Linear Algebra,
• J. Böhm: Grundlagen der Algebra und Zahlentheorie,
• B. Kreußler, G. Pfister: Mathematik für Informatiker: Algebra, Analysis, Diskrete Strukturen,
• V. Shoup: A Computational Introduction to Number Theory and Algebra.

## Registration

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

none

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## References to Module / Module Number [MAT-02-11-M-1]

Course of Study Section Choice/Obligation
[INF-82.79-SG] B.Sc. Computer Science [Compulsory Modules] Theoretical Foundations [P] Compulsory