Groups: finite groups, free groups, presentations with generators and relations.
Categories: category and functor, natural transformations, universal constructions.
Modules: submodules, quotient modules, homomorphisms, exactness, free and projective modules, tensor product of modules.
Lattices: basic properties and examples, special classes of lattices.
Fields: Galois group, Galois correspondence, solvability of polynomial equations by radicals, fundamental theorem of algebra.
Algebra 3
2023/2024
Programme:
Financial mathematics, First Cycle
Year:
3 year
Semester:
second
Kind:
optional
Group:
B
ECTS:
5
Language:
slovenian
Course director:
Lecturer (contact person):
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)
Readings
Vidav: Algebra, DMFA-založništvo, Ljubljana, 2003.
J. Gallian: Contemporary Abstract Algebra, Brooks/Cole, 2013.
P. M. Cohn: Algebra, 2nd edition, John Wiley & Sons, New York, 1997.
T. W. Hungerford: Algebra, Springer, New York-Berlin, 2003.
J. Rotman: Galois Theory, 2nd edition, Springer, New York, 2001.
Objectives and competences
Basic notions in algebra are introduced, which are needed for the subsequent study. Abstract thinking and mathematical rigour are enhanced.
Practical, working knowledge is obtained during exercise classes.
Intended learning outcomes
Knowledge and understanding: Knowledge and understanding of basic algebraic concepts.
Application: Application of the theory in solving problems.
Reflection: Understanding of the theory from the applications.
Transferable skills: Ability to transfer the theory into practice.
Learning and teaching methods
Lectures, exercises, consultations
Assessment
2 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
Primož Moravec:
DELIZIA, Constantino, MORAVEC, Primož, NICOTERA, Chiara. Groups with all centralizers subnormal of defect at most two. Journal of algebra, ISSN 0021-8693, 2013, vol. 374, str. 132-140.[COBISS-SI-ID 16556889]
MORAVEC, Primož. Unramified Brauer groups of finite and infinite groups. American journal of mathematics, ISSN 0002-9327, 2012, vol. 134, no. 6, str. 1679-1704. [COBISS-SI-ID 16521305]
MORAVEC, Primož. Groups of order p [sup] 5 and their unramified Brauer groups. Journal of algebra, ISSN 0021-8693, 2012, vol. 372, str. 420-427. [COBISS-SI-ID 16521049]
Igor Klep:
Derksen, Harm, KLEP, Igor, Makam, Visu, Volčič, Jurij. Ranks of linear matrix pencils separate simultaneous similarity orbits. Advances in mathematics. - ISSN 0001-8708 (Vol. 415, Feb. 2023, art. 108888 (20 str.)) [COBISS-SI-ID 141160195]
Helton, J. William, KLEP, Igor, Volčič, Jurij. Factorization of noncommutative polynomials and Nullstellensätze for the free algebra. International mathematics research notices. - ISSN 1073-7928 (Vol. 2022, iss. 1, Jan. 2022, str. 343-372) [COBISS-SI-ID 48375811]
KLEP, Igor, Pascoe, James E., Volčič, Jurij. Positive univariate trace polynomials. Journal of algebra. - ISSN 0021-8693 (Vol. 579, Aug. 2021, str. 303-317) [COBISS-SI-ID 105932803]
