Algebra I - NALG026

• Title: Algebra I
• Guaranteed by: Department of Algebra (32-KA)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2018
• Semester: winter
• E-Credits: 6
• Hours per week, examination: winter s.:2/2, C+Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: cancelled
• Language: Czech
• Teaching methods: full-time
• Guarantor: prof. RNDr. Jan Trlifaj, CSc., DSc.
• Classification: Mathematics > Algebra
• Pre-requisite : {Linear Algebra and Geometry}
• Interchangeability : NALG034, NALG087, NMAG201, NMAI062
• Is co-requisite for: NALG027
• Is incompatible with: NMUE004, NUMZ004, NMAX062, NMAI062, NUMP019, NUMP007, NMUE033, NMAG201
• Is pre-requisite for: NALG019, NALG009, NALG008
• Is interchangeable with: NALG087, NUMP019, NUMZ004, NMAG201, NMUE033

Annotation

English

Basic concepts and results of group theory. Introduction to rings and modules. Categories and localization.

Last update: T_KA (20.05.2010)

Czech

Základní pojmy a věty z teorie grup. Úvod do okruhů, modulů, lokalizace a kategorií.

Last update: T_KA (19.05.2010)

Literature

English

S.Lang, Algebra, Revised 3rd ed., GTM 211, Springer, New York, 2002.

N. Lauritzen, Concrete Abstract Algebra, Cambridge Univ. Press, Cambridge 2003.

C. Menini and F. van Oystaeyen, Abstract Algebra, M. Dekker, New York 2004.

L.Procházka a kol., Algebra, Academia, Praha, 1990 (in Czech).

J.Trlifaj: Algebra I, http://www.karlin.mff.cuni.cz/~trlifaj/NALG026.pdf (in Czech).

Last update: T_KA (19.05.2010)

Czech

S.Lang, Algebra, Revised 3rd ed., GTM 211, Springer, New York, 2002.

N. Lauritzen, Concrete Abstract Algebra, Cambridge Univ. Press, Cambridge 2003.

C. Menini a F. van Oystaeyen, Abstract Algebra, M. Dekker, New York 2004.

L.Procházka a kol., Algebra, Academia, Praha, 1990.

J.Trlifaj: Algebra I, http://www.karlin.mff.cuni.cz/~trlifaj/NALG026.pdf

Last update: T_KA (19.05.2010)

Syllabus

English

1. Groups and their representations.

1.1 Monoids, The Cayley Theorem.

1.2 Groups, cosets, The Lagrange Theorem.

1.3 Normal subgroups, Noether's isomorphism theorems.

1.4 Cyclic groups, permutation and matrix groups.

1.5 Groups acting on sets; structure of finite abelian groups.

Supplementary topic: Group representations, construction of the regular representation.

2. Rings and localization.

2.1 Ideals and homomorphisms.

2.2 Commutative rings, prime ideals, and localization.

3. Modules and categories.

3.1 Introduction to category theory.

3.2 Module categories, diagrams.

Supplementary topic: completeness of Mod-R.

Last update: T_KA (19.05.2010)

Czech

1. Grupy a reprezentace grup.

1.1 Monoidy, Cayleyho věta.

1.2 Grupy, rozklady podle podgrup, Lagrangeova věta.

1.3 Normální podgrupy, věty o homomorfismu a izomorfismu.

1.4 Cyklické grupy, permutační a maticové grupy.

1.5 Akce grupy na množině; struktura konečných komutativních grup.

Rozšiřující téma: Úvod do teorie reprezentací grup, konstrukce regulární reprezentace.

2. Okruhy a lokalizace.

2.1 Ideály, věty o homomorfismu a izomorfismu.

2.2 Komutativní okruhy, prvoideály a lokalizace.

3. Moduly a kategorie.

3.1 Úvod do teorie kategorií.

3.2 Kategorie modulů, diagramy.

Rozšiřující téma: úplnost Mod-R.

Last update: T_KA (19.05.2010)