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Course, academic year 2023/2024
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Algebra 1 - NMAX062
Title: Algebra 1
Guaranteed by: Student Affairs Department (32-STUD)
Faculty: Faculty of Mathematics and Physics
Actual: from 2021
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: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Is provided by: NMAI062
Additional information:
Guarantor: RNDr. Zuzana Patáková, Ph.D.
Liran Shaul, Ph.D.
Class: Informatika Bc.
Classification: Mathematics > Algebra
Pre-requisite : {NXXX015, NXXX018, NXXX022, NXXX023, NXXX024, NXXX025, NXXX030, NXXX031, NXXX033}
Incompatibility : NALG026, NMAI062
Interchangeability : NMAI062
Is incompatible with: NMAI062
Is interchangeable with: NMAI062
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: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (16.10.2023)

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

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

Lecture notes:

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

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

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:

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