Algebra for computer science

Lecturer:

Alena Gollová

Tutor:

Alena Gollová

Supervisor:

Department of Mathematics

Synopsis:

The course covers selected topics of discrete mathematics for computer

science, namely basic group theory, rings, finite fields and

lattices.

A structure of finite groups, namely of cyclic groups, will be studied and equations x^k=1 will be solved in finite groups.

Students will get acquinted with construction of finite fields and cyclic codes.

Finally distributive lattices and homomorphisms of algebraic

structures will be investigated.

Requirements:

The requirement for receiving the credit is an active participation in the tutorials.

Syllabus of lectures:

1. Groups, Euler-Fermat Theorem. Chinese Remainder Theorem, applications.

2. Subgroups, orders of elements in a finite group.

3. Cyclic groups and their properties. Solving of equations in cyclic groups.

4. Rings and fields of congruence classes.

5. Polynomials over Zp, irreducible polynomials.

6. Euclid's algorithm for polynomials, rings of polynomials

7. Linear and cyclic codes.

8. Fields GF(p^n).

9. Characteristics of a field, primitive element, applications of finite fields.

10. Lattices and partial order.

11. Distributive lattices.

12. Homomorphisms of structures given by operations/relations.

13. Identities and free objects.

Syllabus of tutorials:

1. Groups, Euler-Fermat Theorem. Chinese Remainder Theorem, applications.

2. Subgroups, order of an element in a finite group.

3. Cyclic groups and their properties. Solving of equations in a cyclic group.

4. Rings and fields of congruence classes.

5. Polynomials over Zp, irreducible polynomials.

6. Euclid's algorithm for polynomials, rings of polynomials

7. Linear and cyclic codes.

8. Fields GF(pn).

9. Characteristics of a field, primitive element, applications of finite fields.

10. Lattices and partial order.

11. Distributive lattices.

12. Homomorphisms of structures given by operations/relations.

13. Identities and free objects.

Study Objective:

Study materials:

1. Lindsay Childs: A Concrete Introduction to Higher Algebra, Springer-Verlag, 1979

2. F. P. Preparata, R. T. Yeh: Ontroduction to Discrete Structures, Wesley Publishing Company, Reading USA 1974

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue	roomT2:A4-202a Gollová A. 10:00–11:45 EVEN WEEK (lecture parallel1) Dejvice Ucebna
Fri
Thu
Fri

The course is a part of the following study plans:

• Computer Technology - Software Engineering- structured studies (compulsory course)
• Computer Technology - System Programming- structured studies (compulsory course)
• Computer Technology - Computer Graphics- structured studies (compulsory course)
• Computer Technology - Computer Network and Internet- structured studies (compulsory course)
• Computer Technology - Designing Digital Systems- structured studies (compulsory course)