SubjectsSubjects(version: 875)
Course, academic year 2020/2021
Algebra 1 - NMAI062
Title: Algebra 1
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Additional information:
Guarantor: doc. RNDr. David Stanovský, Ph.D.
Liran Shaul, Ph.D.
Class: Informatika Bc.
Classification: Mathematics > Algebra
Incompatibility : NMAX062
Interchangeability : NMAX062
K//Is co-requisite for: NMAI063
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: Mgr. Jan Šaroch, Ph.D. (12.10.2017)

To obtain "Zápočet", the student has to have an amount of 60 points. These can be gained either from two written tests (maximal 50 points per each), or by correctly solving homework problems (here, the point rating varies).

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: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (09.10.2017)

The exam will be oral. It will consists of three questions; one general asking to overview certain topic, for example, symmetric groups or unique factorization domains, one more specific, for example, to formulate and prove the Lagrange theorem for groups, and one testing the understanding, for example, to find a non-trivial normal subgroup of the group A4 . Students will get time to prepare the answers.

Syllabus -
Last update: T_KA (17.05.2010)

1. Monoids, groups and subgroups. Factorization of groups and normal subgroups.

2. Cyclic groups and RSA.

3. Basic notions of universal algebra: algebra, homomorphism, congruence.

4. Lattices and Boolean algebras.

5. Rings and ideals. Fields of fractions. Construction of finite fields.

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