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Course, academic year 2022/2023
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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
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 -
Last update: T_KA (20.05.2010)
Basic concepts and results of group theory. Introduction to rings and modules. Categories and localization.
Literature -
Last update: T_KA (19.05.2010)

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, (in Czech).

Syllabus -
Last update: T_KA (19.05.2010)

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.

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