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Algebra 1 - NMAI062

Title:	Algebra 1
Guaranteed by:	Department of Algebra (32-KA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2022
Semester:	winter
E-Credits:	5
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:	taught
Language:	Czech, English
Teaching methods:	full-time
Additional information:	https://dl1.cuni.cz/course/view.php?id=14058

Guarantor:	RNDr. Zuzana Patáková, Ph.D. Liran Shaul, Ph.D.
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 Examination dates WS schedule Noticeboard

Annotation -

Last update: T_KA (20.05.2009)

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.

Course completion requirements -

Last update: Michael Kompatscher, Ph.D. (08.10.2021)

To obtain "Zápočet", the student has to have an amount of 60 points. These can be gained either from 3 homeworks (maximal 30 points per each), or by correctly solving weekly quizzes (10 points in total).

Literature -

Last update: T_KA (17.05.2010)

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

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

Requirements to the exam -

Last update: Liran Shaul, Ph.D. (25.09.2020)

The course will be finished with a written exam.

Syllabus -

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

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