Algebra 2

2022/2023
Programme:
Mathematics, First Cycle
Year:
2 year
Semester:
first and second
Kind:
mandatory
ECTS:
10
Language:
slovenian
Hours per week – 1. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed course Algebra 1.

Content (Syllabus outline)

Groups: binary operations, semigroups, monoids, and groups: basic properties and examples, subgroups and cosets, homomorphisms, normal subgroups and quotient groups, introduction to the structure theory of groups, finite abelian groups.
Rings: basic properties and examples, homomorphisms, ideals and quotients rings, field of fractions, principal ideal domains, rings of polynomials in one and several variables.
Fields: finite extensions, algebraic and transcendental elements and extensions, constructible numbers, splitting fields, algebraically closed fields, finite fields.

Readings

Vidav: Algebra, DMFA-založništvo, Ljubljana, 2003.
J. Gallian: Contemporary Abstract Algebra, Brooks/Cole, 2013.
I. N. Herstein: Abstract Algebra, John Wiley & Sons, 1999.
S. Lang: Undergraduate Algebra, Springer, 2005.
L. H. Rowen: Algebra. Groups, rings, and fields. A K Peters, 1994.

Objectives and competences

Students learn basic concepts of algebra that are needed for further study of mathematics. At the same time they learn abstract thinking and rigorous mathematical language. In the tutorial they acquire practical, working knowledge of the area.

Intended learning outcomes

Knowledge and understanding: Knowledge and understanding of basic algebraic concepts.
Application: Application of the theory to problem solving.
Reflection: Understanding the theory through applications.
Transferable skills: Transfer of theory into practice.

Learning and teaching methods

Lectures, exercises, consultations

Assessment

3 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

BREŠAR, Matej. Introduction to noncommutative algebra, (Universitext). Cham [etc.]: Springer, cop. 2014. XXXVII, 199 str. ISBN 978-3-319-08692-7. ISBN 978-3-319-08693-4. [COBISS-SI-ID 17143897]
BREŠAR, Matej, ŠPENKO, Špela. Functional identities of one variable. Journal of algebra, ISSN 0021-8693, 2014, vol. 401, str. 234-244. [COBISS-SI-ID 16842329]
BREŠAR, Matej, KLEP, Igor. A local-global principle for linear dependence of noncommutative polynomials. Israel journal of mathematics, ISSN 0021-2172, 2013, vol. 193, iss. 1, str. 71-82. [COBISS-SI-ID 16626521]