## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Course MAT-14-12-K-3

## Course Type

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

## Basedata

SWS 4V+2U 9.0 CP = 270 h 1 Sem. in SuSe [3] Bachelor (Core) [DE] German Fieker, Claus, Prof. Dr. (PROF | DEPT: MAT) Malle, Gunter, Prof. Dr. (PROF | DEPT: MAT) Schulze, Mathias, Prof. Dr. (PROF | DEPT: MAT) Thiel, Ulrich, Prof. Dr. (PROF | DEPT: MAT) Böhm, Janko, Dr. (WMA | DEPT: MAT) [MAT-GRU] Mathematics (B.Sc. year 1 and 2) [NORM] Active

## Notice

The course is accompanied by a programming lab course: [MAT-14-12P-K-3].

## Possible Study achievement

• Verification of study performance: proof of successful participation in the exercise classes (ungraded)
• Examination number (Study achievement): 83240 ("Exercise Class Introduction to Symbolic Computing")
• Details of the examination (type, duration, criteria) will be announced at the beginning of the course.

## Contents

• primality testing and factorisation of integers,
• polynomial arithmetic (fast polynomial multiplication, modular gcd computation, factorisation),
• modules over principal ideal domains (structural theorem, Hermite and Smith normal form),
• Gröbner bases for ideals and modules,
• lattices (rational reconstruction, LLL algorithm, application to polynomial factorization).

## Competencies / intended learning achievements

The students are familiar with modern methods of symbolic computing and their complexity. In particular, they have developed a feeling for the design of algebraic algorithms and their practical implementation. The latter is deepened in the optional programming lab course.

## Literature

• H. Cohen: A Course in Computational Algebraic Number Theory,
• D. A. Cox, J. Little, D. O'Shea: Ideals, Varieties, and Algorithms,
• W. Decker, G. Pfister: A First Course in Computational Algebraic Geometry,
• J. von zur Gathen, J. Gerhard: Modern Computer Algebra,
• D. Knuth: The Art of Computer Programming. Volumes 1,2,3,
• R. Lidl, H. Niederreiter: Introduction to Finite Fields and Their Applications,
• G.-M. Greuel, G. Pfister: A SINGULAR Introduction to Commutative Algebra.

## Materials

Further literature will be announced in the lecture(s); exercise material is provided.

## Registration

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

None

## References to Course [MAT-14-12-K-3]

Module Name Context
[MAT-14-10L-M-3] Topic Module B: Mathematics as an Interdisciplinary Cross-Sectional Science WP: Obligation to choose 4V+2U, 9.0 LP
[MAT-14-12-M-3] Introduction to Symbolic Computing P: Obligatory 4V+2U, 9.0 LP
Course-Pool Name
[MAT-14-KPOOL-3] Practical Mathematics (B.Sc. Mathematics)