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Algebra 1 - NMAI062

Title:	Algebra 1
Form of teaching:	lecture+practicals
Guaranteed by:	Department of Algebra (32-KA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2025 to 2025
Duration in semesters:	1
Semester:	winter
E-Credits:	5
Hours per week, examination:	winter s.:2/2, C+Ex [HT]
Capacity:	unlimited
Maximum number of enrolled students:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	English, Czech
Teaching methods:	full-time
Additional information:	https://dl1.cuni.cz/course/view.php?id=14058
Repeated enrollment:	2 / 2 / 2 / 2

Guarantor:	RNDr. Zuzana Patáková, Ph.D. Stefano Fioravanti, Ph.D.
Teacher(s):	Alexey Barsukov, Ph.D. Stefano Fioravanti, Ph.D. Bc. Ondřej Chwiedziuk Yagmur Sak
Class:	Informatika Bc.
Classification:	Mathematics > Algebra
Incompatibility :	NALG026, NMAX062
Interchangeability :	NMAX062
Is co-requisite for:	NMAI063, NMAI076, NMAX063
Is incompatible with:	NUMP019, NMAX062
Is interchangeable with:	NMAX062, NALG034, NALG087, NALG026, NUMP019

Opinion survey results WS schedule Noticeboard

Annotation -

The course in basic algebra is devoted to fundamental algebraic notions that are demonstrated on basic algebraic structures. Notions include closure systems, operations, algebras (as sets with operations), homomorphisms, congruences, orderings and the divisibility. Lattices, monoids, groups, rings and fields are regarded as the basic structures. The course also pays attention to modular arithmetic and finite fields.

Last update: T_KA (20.05.2009)

Course completion requirements -

There will be four homework sets, each worth 25 points. To get credit (zápočet) you need to get 64 points in total.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (16.10.2023)

Literature -

Lecture notes: https://www.karlin.mff.cuni.cz/~kompatscher/teaching/alg1_en.pdf

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

S. MacLane, G. Birkhoff, Algebra 3rd ed, Providence 1999, AMS Chelsea publishing company.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (16.10.2023)

Course assessment methods and requirements for successful completion, grading scheme -

The course will be ended by a written exam followed by an oral exam based on the results of the written one.

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (16.10.2023)

Syllabus -

1) Number theory: prime factorization, congruences, Euler's theorem and RSA, the Chinese remainder theorem

2) Polynomials: rings and integral domains, polynomial rings, irreducibility, GCD, the Chinese remainder theorem and interpolation, the construction of finite fields and applications (error-correcting codes, secret sharing,...)

3) Group theory: permutation groups, subgroups, Lagrange's theorem, group actions and Burnsides's theorem, cyclic groups, discrete logarithm and applications in cryptography

see also: https://www.logic.at/staff/kompatscher/algebra1.html

Last update: Kompatscher Michael, Ph.D. (28.09.2021)