Algebra

Course guarantor:

Zuzana Masáková

Lecturer:

Zuzana Masáková

Tutor:

Zuzana Masáková

Supervisor:

Department of Mathematics

Synopsis:

The lecture discusses some parts of set theory, e.g. equivalence and subvalence of sets, the Cantor-Bernstein theorem, the axiom of choice and equivalent statements. Furthermore, standard algebraic structures are discussed: semigroups, monoids, groups, rings, integral domains, principal ideal domains, fields, lattices. Independent chapters are devoted to divisibility in integral domains and to finite fields.

Requirements:

To obtain credit, active participation in the exercises is required, including demonstrating the solution of several pre-prepared problems at the blackboard.

The exam is oral.

Syllabus of lectures:

1. Binary relations, equivalence, ordering

2. Equivalence and subvalence of sets, the transfinite induction

3. The axiom of choice and equivalent statements

4. Semigroups, monoids

5. Groups

6. Rings, integral domains, principal ideal domains, fields

7. Divisibility in integral domains

8. Finite fields

9. Lattices

Syllabus of tutorials:

Study Objective:

Knowledge: elements of the set theory - equivalence and subvalence, the axiom of choice and equivalent statements; basics of algebra - monoids, groups, rings, integral domains, principal ideal domains, fields

Skills: using algebraic structures, applying these structures along with some elements of the set theory in other fields of mathematics

Study materials:

Key references:

[1] Mareš J.: Algebra. Úvod do obecné algebry, 3. vydání. ČVUT, Praha, 1999.

[2] Šťovíček P.: Úvod do obecné algebry s prvky teorie množin, ČVUT , Praha, 2021.

Recommended references:

[3] Mac Lane S., Birkhoff G.: Algebra. Springer, New York, 2005.

[3] Lang S.: Algebra. Springer, New York, 2005.

Note:

Time-table for winter semester 2024/2025:

	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	roomTR:210 Masáková Z. 14:00–15:50 (lecture parallel1) Trojanova 13
Tue	roomTR:210 Masáková Z. 08:00–09:50 (parallel nr.101) Trojanova 13 roomTR:210 Masáková Z. 10:00–11:50 (lecture parallel1) Trojanova 13
Wed
Thu
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Matematické inženýrství - Matematická fyzika (elective course)
• Matematické inženýrství - Matematická informatika (PS)
• Matematické inženýrství - Matematické modelování (PS)
• Mathematical Engineering - Mathematical Physics (elective course)
• Mathematical Engineering - Mathematical Modelling (PS)
• Mathematical Engineering - Mathematical Informatics (PS)